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-rw-r--r--šola/citati.bib227
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-rw-r--r--šola/članki/dht/dokument.lyx2306
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+#LyX 2.4 created this file. For more info see https://www.lyx.org/
+\lyxformat 620
+\begin_document
+\begin_header
+\save_transient_properties true
+\origin unavailable
+\textclass article
+\begin_preamble
+\usepackage{hyperref}
+\usepackage{siunitx}
+\usepackage{pgfplots}
+\usepackage{listings}
+\usepackage{multicol}
+\sisetup{output-decimal-marker = {,}, quotient-mode=fraction, output-exponent-marker=\ensuremath{\mathrm{3}}}
+\usepackage{amsmath}
+\usepackage{tikz}
+\newcommand{\udensdash}[1]{%
+ \tikz[baseline=(todotted.base)]{
+ \node[inner sep=1pt,outer sep=0pt] (todotted) {#1};
+ \draw[densely dashed] (todotted.south west) -- (todotted.south east);
+ }%
+}%
+\DeclareMathOperator{\Lin}{\mathcal Lin}
+\DeclareMathOperator{\rang}{rang}
+\DeclareMathOperator{\sled}{sled}
+\DeclareMathOperator{\Aut}{Aut}
+\DeclareMathOperator{\red}{red}
+\DeclareMathOperator{\karakteristika}{char}
+\DeclareMathOperator{\Ker}{Ker}
+\DeclareMathOperator{\Slika}{Ker}
+\DeclareMathOperator{\sgn}{sgn}
+\DeclareMathOperator{\End}{End}
+\DeclareMathOperator{\n}{n}
+\DeclareMathOperator{\Col}{Col}
+\usepackage{algorithm,algpseudocode}
+\providecommand{\corollaryname}{Posledica}
+\usepackage[slovenian=quotes]{csquotes}
+\end_preamble
+\use_default_options true
+\begin_modules
+enumitem
+theorems-ams
+theorems-ams-extended
+\end_modules
+\maintain_unincluded_children no
+\language slovene
+\language_package default
+\inputencoding auto-legacy
+\fontencoding auto
+\font_roman "default" "default"
+\font_sans "default" "default"
+\font_typewriter "default" "default"
+\font_math "auto" "auto"
+\font_default_family default
+\use_non_tex_fonts false
+\font_sc false
+\font_roman_osf false
+\font_sans_osf false
+\font_typewriter_osf false
+\font_sf_scale 100 100
+\font_tt_scale 100 100
+\use_microtype false
+\use_dash_ligatures true
+\graphics default
+\default_output_format default
+\output_sync 0
+\bibtex_command default
+\index_command default
+\float_placement class
+\float_alignment class
+\paperfontsize default
+\spacing single
+\use_hyperref false
+\papersize default
+\use_geometry true
+\use_package amsmath 1
+\use_package amssymb 1
+\use_package cancel 1
+\use_package esint 1
+\use_package mathdots 1
+\use_package mathtools 1
+\use_package mhchem 1
+\use_package stackrel 1
+\use_package stmaryrd 1
+\use_package undertilde 1
+\cite_engine basic
+\cite_engine_type default
+\biblio_style plain
+\use_bibtopic false
+\use_indices false
+\paperorientation portrait
+\suppress_date false
+\justification false
+\use_refstyle 1
+\use_formatted_ref 0
+\use_minted 0
+\use_lineno 0
+\index Index
+\shortcut idx
+\color #008000
+\end_index
+\leftmargin 2cm
+\topmargin 2cm
+\rightmargin 2cm
+\bottommargin 2cm
+\headheight 2cm
+\headsep 2cm
+\footskip 1cm
+\secnumdepth 3
+\tocdepth 3
+\paragraph_separation indent
+\paragraph_indentation default
+\is_math_indent 0
+\math_numbering_side default
+\quotes_style german
+\dynamic_quotes 0
+\papercolumns 1
+\papersides 1
+\paperpagestyle default
+\tablestyle default
+\tracking_changes false
+\output_changes false
+\change_bars false
+\postpone_fragile_content false
+\html_math_output 0
+\html_css_as_file 0
+\html_be_strict false
+\docbook_table_output 0
+\docbook_mathml_prefix 1
+\end_header
+
+\begin_body
+
+\begin_layout Title
+Teorija Analize 1 —
+ IŠRM 2023/24
+\end_layout
+
+\begin_layout Author
+
+\noun on
+Anton Luka Šijanec
+\end_layout
+
+\begin_layout Date
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+today
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Abstract
+Povzeto po zapiskih s predavanj profesorja Oliverja Dragičevića.
+\end_layout
+
+\begin_layout Standard
+\begin_inset CommandInset toc
+LatexCommand tableofcontents
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Števila
+\end_layout
+
+\begin_layout Definition*
+Množica je matematični objekt,
+ ki predstavlja skupino elementov.
+ Če element
+\begin_inset Formula $a$
+\end_inset
+
+ pripada množici
+\begin_inset Formula $A$
+\end_inset
+
+,
+ pišemo
+\begin_inset Formula $a\in A$
+\end_inset
+
+,
+ sicer pa
+\begin_inset Formula $a\not\in A$
+\end_inset
+
+.
+ Množica
+\begin_inset Formula $B$
+\end_inset
+
+ je podmnožica množice
+\begin_inset Formula $A$
+\end_inset
+
+,
+ pišemo
+\begin_inset Formula $B\subset A$
+\end_inset
+
+,
+ če
+\begin_inset Formula $\forall b\in B:b\in A$
+\end_inset
+
+.
+ Presek
+\begin_inset Formula $B$
+\end_inset
+
+ in
+\begin_inset Formula $C$
+\end_inset
+
+ označimo
+\begin_inset Formula $B\cap C\coloneqq\left\{ x;x\in B\wedge x\in C\right\} $
+\end_inset
+
+.
+ Unijo
+\begin_inset Formula $B$
+\end_inset
+
+ in
+\begin_inset Formula $C$
+\end_inset
+
+ označimo
+\begin_inset Formula $B\cup C\coloneqq\left\{ x;x\in B\vee x\in C\right\} $
+\end_inset
+
+.
+ Razliko/komplement
+\begin_inset Quotes gld
+\end_inset
+
+
+\begin_inset Formula $B$
+\end_inset
+
+ manj/brez
+\begin_inset Formula $C$
+\end_inset
+
+
+\begin_inset Quotes grd
+\end_inset
+
+ označimo
+\begin_inset Formula $B\setminus C\coloneqq\left\{ x;x\in B\wedge x\not\in C\right\} $
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Subsection
+Realna števila
+\end_layout
+
+\begin_layout Standard
+Množico realnih števil označimo
+\begin_inset Formula $\mathbb{R}$
+\end_inset
+
+.
+ V njej obstajata binarni operaciji seštevanje
+\begin_inset Formula $a+b$
+\end_inset
+
+ in množenje
+\begin_inset Formula $a\cdot b$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Subsubsection
+Lastnosti seštevanja
+\end_layout
+
+\begin_layout Axiom
+Komutativnost:
+
+\begin_inset Formula $\forall a,b\in\mathbb{R}:a+b=b+a$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Separator plain
+\end_inset
+
+
+\end_layout
+
+\begin_layout Axiom
+Asociativnost:
+
+\begin_inset Formula $\forall a,b,c\in\mathbb{R}:a+\left(b+c\right)=\left(a+b\right)+c$
+\end_inset
+
+,
+ torej je
+\begin_inset Formula $a+\cdots+z$
+\end_inset
+
+ dobro definiran izraz.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Separator plain
+\end_inset
+
+
+\end_layout
+
+\begin_layout Axiom
+Obstoj enote:
+
+\begin_inset Formula $\exists0\in\mathbb{R}\forall a\in\mathbb{R}:a+0=a$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Separator plain
+\end_inset
+
+
+\end_layout
+
+\begin_layout Axiom
+Obstoj inverzov:
+
+\begin_inset Formula $\forall a\in\mathbb{R}\exists b\in\mathbb{R}\ni a+b=0$
+\end_inset
+
+.
+\end_layout
+
+\begin_deeper
+\begin_layout Claim*
+Inverz je enoličen.
+\end_layout
+
+\begin_layout Proof
+Naj bo
+\begin_inset Formula $a,b,c\in\mathbb{R}$
+\end_inset
+
+ in
+\begin_inset Formula $a+b=0$
+\end_inset
+
+ in
+\begin_inset Formula $a+c=0$
+\end_inset
+
+.
+ Tedaj
+\begin_inset Formula $b=b+0=b+a+c=0+c=c$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Corollary*
+Inverz je funkcija in aditivni inverz
+\begin_inset Formula $a$
+\end_inset
+
+ označimo z
+\begin_inset Formula $-a$
+\end_inset
+
+.
+ Pri zapisu
+\begin_inset Formula $a+\left(-b\right)$
+\end_inset
+
+ običajno
+\begin_inset Formula $+$
+\end_inset
+
+ izpustimo in pišemo
+\begin_inset Formula $a-b$
+\end_inset
+
+,
+ čemur pravimo odštevanje
+\begin_inset Formula $b$
+\end_inset
+
+ od
+\begin_inset Formula $a$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Claim*
+\begin_inset Formula $\forall a\in\mathbb{R}:a=-\left(-a\right)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Proof
+Naj bo
+\begin_inset Formula $b=-a$
+\end_inset
+
+ in
+\begin_inset Formula $c=-b=-\left(-a\right)$
+\end_inset
+
+.
+ Tedaj velja
+\begin_inset Formula $c-a=c+b=-\left(-a\right)-a=0$
+\end_inset
+
+ in
+\begin_inset Formula $a=0+a=c-a+a=c+\left(-a\right)+a=c+0=c=-\left(-a\right)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Claim*
+\begin_inset Formula $-\left(b+c\right)=-b-c$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Proof
+Velja
+\begin_inset Formula $b+c+\left(-b-c\right)=b+c+\left(\left(-b\right)+\left(-c\right)\right)=b+\left(-b\right)+c+\left(-c\right)=0$
+\end_inset
+
+,
+ torej je
+\begin_inset Formula $b+c$
+\end_inset
+
+ inverz od
+\begin_inset Formula $\left(-b-c\right)$
+\end_inset
+
+,
+ torej je
+\begin_inset Formula $-\left(b+c\right)=-b-c$
+\end_inset
+
+.
+\end_layout
+
+\end_deeper
+\begin_layout Subsubsection
+Lastnosti množenja
+\end_layout
+
+\begin_layout Axiom
+Komutativnost:
+
+\begin_inset Formula $\forall a,b\in\mathbb{R}:ab=ba$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Separator plain
+\end_inset
+
+
+\end_layout
+
+\begin_layout Axiom
+Asociativnost:
+
+\begin_inset Formula $\forall a,b,c\in\mathbb{R}:a\left(bc\right)=\left(ab\right)c$
+\end_inset
+
+,
+ torej je
+\begin_inset Formula $a\cdots z$
+\end_inset
+
+ dobro definiran izraz.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Separator plain
+\end_inset
+
+
+\end_layout
+
+\begin_layout Axiom
+Obstoj enote:
+
+\begin_inset Formula $\exists1\in\mathbb{R}\forall a\in\mathbb{R}:a1=a$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Separator plain
+\end_inset
+
+
+\end_layout
+
+\begin_layout Axiom
+Obstoj inverzov:
+
+\begin_inset Formula $\forall a\in\mathbb{R}\setminus\left\{ 0\right\} \exists b\in\mathbb{R}\setminus\left\{ 0\right\} \ni:ab=1$
+\end_inset
+
+
+\end_layout
+
+\begin_deeper
+\begin_layout Claim*
+Inverz je enoličen.
+\end_layout
+
+\begin_layout Proof
+Naj bo
+\begin_inset Formula $a,b,c\in\mathbb{R}\setminus\left\{ 0\right\} $
+\end_inset
+
+ in
+\begin_inset Formula $ab=1$
+\end_inset
+
+ in
+\begin_inset Formula $ac=1$
+\end_inset
+
+.
+ Tedaj
+\begin_inset Formula $b=b1=bac=1c=c$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Corollary*
+Inverz je funkcija in multiplikativni inverz
+\begin_inset Formula $a$
+\end_inset
+
+ označimo z
+\begin_inset Formula $a^{-1}$
+\end_inset
+
+.
+ Pri zapisu
+\begin_inset Formula $a\cdot b^{-1}$
+\end_inset
+
+ lahko
+\begin_inset Formula $\cdot$
+\end_inset
+
+ izpustimo in pišemo
+\begin_inset Formula $a/b$
+\end_inset
+
+,
+ čemur pravimo deljenje
+\begin_inset Formula $a$
+\end_inset
+
+ z
+\begin_inset Formula $b$
+\end_inset
+
+ za neničeln
+\begin_inset Formula $b$
+\end_inset
+
+.
+\end_layout
+
+\end_deeper
+\begin_layout Subsubsection
+Skupne lastnosti v
+\begin_inset Formula $\mathbb{R}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Axiom
+\begin_inset Formula $1\not=0$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Separator plain
+\end_inset
+
+
+\end_layout
+
+\begin_layout Axiom
+Distributivnost:
+
+\begin_inset Formula $\forall a,b,c\in\mathbb{R}:\left(a+b\right)c=ac+bc$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Paragraph
+Urejenost
+\begin_inset Formula $\mathbb{R}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Realna števila delimo na pozitivna
+\begin_inset Formula $\mathbb{R}_{+}\coloneqq\left\{ x\in\mathbb{R};x>0\right\} $
+\end_inset
+
+,
+ negativna
+\begin_inset Formula $\mathbb{R}_{-}\coloneqq\left\{ x\in\mathbb{R};x<0\right\} $
+\end_inset
+
+ in ničlo
+\begin_inset Formula $0$
+\end_inset
+
+.
+ Če je
+\begin_inset Formula $x\in\mathbb{\mathbb{R}}_{+}\cup\left\{ 0\right\} $
+\end_inset
+
+,
+ pišemo
+\begin_inset Formula $x\geq0$
+\end_inset
+
+,
+ če je
+\begin_inset Formula $x\in\mathbb{R}_{-}\cup\left\{ 0\right\} $
+\end_inset
+
+,
+ pišemo
+\begin_inset Formula $x\leq0$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Axiom
+Če je
+\begin_inset Formula $a\not=0$
+\end_inset
+
+,
+ je natanko eno izmed
+\begin_inset Formula $\left\{ a,-a\right\} $
+\end_inset
+
+ pozitivno,
+ imenujemo ga absolutna vrednost
+\begin_inset Formula $a$
+\end_inset
+
+ (pišemo
+\begin_inset Formula $\left|a\right|$
+\end_inset
+
+),
+ in natanko eno negativno,
+ pišemo
+\begin_inset Formula $-\left|a\right|$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Definition*
+\begin_inset Formula $\left|0\right|=0$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Separator plain
+\end_inset
+
+
+\end_layout
+
+\begin_layout Definition*
+Za
+\begin_inset Formula $a,b\in\mathbb{R}$
+\end_inset
+
+ se
+\begin_inset Formula $\left|a-b\right|$
+\end_inset
+
+ imenuje razdalja.
+\end_layout
+
+\begin_layout Axiom
+\begin_inset Formula $\forall a,b\in\mathbb{R}:a,b>0\Rightarrow\left(a+b>0\right)\wedge\left(ab>0\right)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Definition*
+Za
+\begin_inset Formula $a,b\in\mathbb{R}$
+\end_inset
+
+:
+
+\begin_inset Formula $a$
+\end_inset
+
+ je večje od
+\begin_inset Formula $b$
+\end_inset
+
+,
+ oznaka
+\begin_inset Formula $a>b\Leftrightarrow a-b>0$
+\end_inset
+
+.
+
+\begin_inset Formula $a$
+\end_inset
+
+ je manjše od
+\begin_inset Formula $b$
+\end_inset
+
+,
+ oznaka
+\begin_inset Formula $a<b\Leftrightarrow a-b<0$
+\end_inset
+
+.
+ Podobno
+\begin_inset Formula $\leq$
+\end_inset
+
+ in
+\begin_inset Formula $\geq$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Claim*
+Trikotniška neenakost.
+
+\begin_inset Formula $\forall a,b\in\mathbb{R}$
+\end_inset
+
+:
+
+\begin_inset Formula $\left|\left|a\right|-\left|b\right|\right|\leq\left|a+b\right|\leq\left|a\right|+\left|b\right|$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Proof
+Dokažimo desni neenačaj.
+ Vemo
+\begin_inset Formula $ab\leq\left|ab\right|$
+\end_inset
+
+ in
+\begin_inset Formula $\left|a\right|=\sqrt{a^{2}}$
+\end_inset
+
+.
+ Naj bo
+\begin_inset Formula $a^{2}+2ab+b^{2}\leq\left|a\right|^{2}+2\left|a\right|\left|b\right|+\left|b\right|^{2}$
+\end_inset
+
+,
+ torej
+\begin_inset Formula $\left(a+b\right)^{2}\leq\left(\left|a\right|+\left|b\right|\right)^{2}$
+\end_inset
+
+,
+ korenimo:
+
+\begin_inset Formula $\left|a+b\right|\leq\left|a\right|+\left|b\right|$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Subsubsection
+Intervali
+\end_layout
+
+\begin_layout Definition*
+Naj bo
+\begin_inset Formula $a<b$
+\end_inset
+
+.
+ Označimo odprti interval
+\begin_inset Formula $\left(a,b\right)\coloneqq\left\{ x\in\mathbb{R};a<x<b\right\} $
+\end_inset
+
+,
+ zaprti
+\begin_inset Formula $\left[a,b\right]\coloneqq\left\{ x\in\mathbb{R};a\leq x\leq b\right\} $
+\end_inset
+
+,
+ polodprti
+\begin_inset Formula $(a,b]\coloneqq\left\{ x\in\mathbb{R};a<x\leq b\right\} $
+\end_inset
+
+ in podobno
+\begin_inset Formula $[a,b)$
+\end_inset
+
+.
+
+\begin_inset Formula $\left(a,\infty\right)\coloneqq\left\{ x\in\mathbb{R};x>a\right\} $
+\end_inset
+
+ in podobno
+\begin_inset Formula $[a,\infty)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Subsection
+Temeljne številske podmnožice
+\end_layout
+
+\begin_layout Subsubsection
+Naravna števila
+\begin_inset Formula $\mathbb{N}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Definition*
+\begin_inset Formula $\mathbb{N}\coloneqq\left\{ 1,1+1,1+1+1,1+1+1+1,\dots\right\} $
+\end_inset
+
+
+\end_layout
+
+\begin_layout Paragraph
+Matematična indukcija
+\end_layout
+
+\begin_layout Standard
+Če je
+\begin_inset Formula $A\subseteq\mathbb{N}$
+\end_inset
+
+ in velja
+\begin_inset Formula $1\in A$
+\end_inset
+
+ (baza) in
+\begin_inset Formula $a\in A\Rightarrow a+1\in A$
+\end_inset
+
+ (korak),
+ tedaj
+\begin_inset Formula $A=\mathbb{N}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Claim*
+\begin_inset Formula $1+2+3+\cdots+n=\frac{n\left(n+1\right)}{2}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Proof
+\begin_inset Formula $A\coloneqq\left\{ n\in\mathbb{N};\text{velja trditev za }n\right\} $
+\end_inset
+
+.
+ Dokažimo
+\begin_inset Formula $A=\mathbb{N}$
+\end_inset
+
+.
+\end_layout
+
+\begin_deeper
+\begin_layout Itemize
+Baza:
+
+\begin_inset Formula $1=\frac{1\cdot2}{2}=1$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+Korak:
+ Predpostavimo
+\begin_inset Formula $1+2+3+\cdots+n=\frac{n\left(n+1\right)}{2}$
+\end_inset
+
+.
+ Prištejmo
+\begin_inset Formula $n+1$
+\end_inset
+
+:
+
+\begin_inset Formula
+\[
+1+2+3+\cdots+n+\left(n+1\right)=\frac{n\left(n+1\right)}{2}+\left(n+1\right)=\frac{n\left(n+1\right)}{2}+\frac{2\left(n+1\right)}{2}=\frac{\left(n+2\right)\left(n+1\right)}{2}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\end_deeper
+\begin_layout Subsubsection
+Cela števila
+\begin_inset Formula $\mathbb{Z}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Množica
+\begin_inset Formula $\mathbb{N}$
+\end_inset
+
+ je zaprta za seštevanje in množenje,
+ torej
+\begin_inset Formula $\forall a,b\in\mathbb{N}:a+b\in\mathbb{N}\wedge ab\in\mathbb{N}$
+\end_inset
+
+,
+ ni pa zaprta za odštevanje,
+ ker recimo
+\begin_inset Formula $5-3\not\in\mathbb{N}$
+\end_inset
+
+.
+ Zapremo jo za odštevanje in dobimo množico
+\begin_inset Formula $\mathbb{Z}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Definition*
+\begin_inset Formula $\mathbb{Z}\coloneqq\left\{ a-b;b,a\in\mathbb{N}\right\} $
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsubsection
+Racionalna števila
+\begin_inset Formula $\mathbb{Q}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Najmanjša podmnožica
+\begin_inset Formula $\mathbb{R}$
+\end_inset
+
+,
+ ki vsebuje
+\begin_inset Formula $\mathbb{Z}$
+\end_inset
+
+ in je zaprta za deljenje,
+ je
+\begin_inset Formula $\mathbb{Q}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Definition*
+\begin_inset Formula $\mathbb{Q}\coloneqq\left\{ a/b;a\in\mathbb{Z},b\in\mathbb{Z}\setminus\left\{ 0\right\} \right\} $
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Velja
+\begin_inset Formula $\mathbb{N}\subset\mathbb{Z}\subset\mathbb{Q}\subset\mathbb{R}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Claim*
+Za
+\begin_inset Formula $a\in\mathbb{Q}$
+\end_inset
+
+,
+
+\begin_inset Formula $b\not\in\mathbb{Q}$
+\end_inset
+
+ velja
+\begin_inset Formula $a+b\not\in\mathbb{Q}$
+\end_inset
+
+ in
+\begin_inset Formula $a\not=0\Rightarrow ab\not\in\mathbb{Q}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Proof
+PDDRAA
+\begin_inset Formula $a+b\in\mathbb{Q}$
+\end_inset
+
+.
+ Tedaj
+\begin_inset Formula $a+b-a\in\mathbb{Q}$
+\end_inset
+
+,
+ tedaj
+\begin_inset Formula $b\in\mathbb{Q}$
+\end_inset
+
+,
+ kar je
+\begin_inset Formula $\rightarrow\!\leftarrow$
+\end_inset
+
+.
+ PDDRAA
+\begin_inset Formula $ab\in\mathbb{Q}$
+\end_inset
+
+.
+ Tedaj
+\begin_inset Formula $\frac{ab}{a}\in\mathbb{Q}$
+\end_inset
+
+,
+ tedaj
+\begin_inset Formula $b\in\mathbb{Q}$
+\end_inset
+
+,
+ kar je
+\begin_inset Formula $\rightarrow\!\leftarrow$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Subsection
+\begin_inset CommandInset label
+LatexCommand label
+name "subsec:Omejenost-množic"
+
+\end_inset
+
+Omejenost množic
+\end_layout
+
+\begin_layout Definition*
+Naj bo
+\begin_inset Formula $A\subset\mathbb{R}$
+\end_inset
+
+.
+
+\begin_inset Formula $A$
+\end_inset
+
+ je navzgor omejena
+\begin_inset Formula $\Leftrightarrow\exists m\in\mathbb{R}\forall a\in A:a\leq m$
+\end_inset
+
+.
+ Takemu
+\begin_inset Formula $m$
+\end_inset
+
+ pravimo zgornja meja.
+ Najmanjši zgornji meji
+\begin_inset Formula $A$
+\end_inset
+
+ pravimo supremum ali natančna zgornja meja množice
+\begin_inset Formula $A$
+\end_inset
+
+,
+ označimo
+\begin_inset Formula $\sup A$
+\end_inset
+
+.
+ Če je zgornja meja
+\begin_inset Formula $A$
+\end_inset
+
+ (
+\begin_inset Formula $m$
+\end_inset
+
+) element
+\begin_inset Formula $A$
+\end_inset
+
+,
+ je maksimum množice
+\begin_inset Formula $A$
+\end_inset
+
+,
+ označimo
+\begin_inset Formula $m=\max A$
+\end_inset
+
+.
+ Če množica ni navzgor omejena,
+ pišemo
+\begin_inset Formula $\sup A=\infty$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Če
+\begin_inset Formula $s=\sup A\in\mathbb{R}$
+\end_inset
+
+,
+ mora veljati
+\begin_inset Formula $\forall a\in A:a\leq s$
+\end_inset
+
+ in
+\begin_inset Formula $\forall\varepsilon>0\exists b\in A\ni:b>s-\varepsilon$
+\end_inset
+
+,
+ torej za vsak neničeln
+\begin_inset Formula $\varepsilon$
+\end_inset
+
+
+\begin_inset Formula $s-\varepsilon$
+\end_inset
+
+ ni več natančna zgornja meja za
+\begin_inset Formula $A$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Definition*
+Naj bo
+\begin_inset Formula $A\subset\mathbb{R}$
+\end_inset
+
+.
+
+\begin_inset Formula $A$
+\end_inset
+
+ je navzdol omejena
+\begin_inset Formula $\Leftrightarrow\exists m\in\mathbb{R}\forall a\in A:a\geq m$
+\end_inset
+
+.
+ Takemu
+\begin_inset Formula $m$
+\end_inset
+
+ pravimo spodnja meja.
+ Največji spodnji meji
+\begin_inset Formula $A$
+\end_inset
+
+ pravimo infimum ali natančna spodnja meja množice
+\begin_inset Formula $A$
+\end_inset
+
+,
+ označimo
+\begin_inset Formula $\inf A$
+\end_inset
+
+.
+ Če je spodnja meja
+\begin_inset Formula $A$
+\end_inset
+
+ (
+\begin_inset Formula $m$
+\end_inset
+
+) element
+\begin_inset Formula $A$
+\end_inset
+
+,
+ je minimum množice
+\begin_inset Formula $A$
+\end_inset
+
+,
+ označimo
+\begin_inset Formula $m=\min A$
+\end_inset
+
+.
+ Če množica ni navzdol omejena,
+ pišemo
+\begin_inset Formula $\inf A=-\infty$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Separator plain
+\end_inset
+
+
+\end_layout
+
+\begin_layout Definition*
+Množica
+\begin_inset Formula $A\subset\mathbb{R}$
+\end_inset
+
+ je omejena,
+ če je hkrati navzgor in navzdol omejena.
+\end_layout
+
+\begin_layout Axiom
+\begin_inset CommandInset label
+LatexCommand label
+name "axm:Dedekind.-Vsaka-navzgor"
+
+\end_inset
+
+Dedekind.
+ Vsaka navzgor omejena množica v
+\begin_inset Formula $\mathbb{R}$
+\end_inset
+
+ ima natančno zgornjo mejo v
+\begin_inset Formula $\mathbb{R}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Remark*
+Za
+\begin_inset Formula $\mathbb{Q}$
+\end_inset
+
+ aksiom
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "axm:Dedekind.-Vsaka-navzgor"
+plural "false"
+caps "false"
+noprefix "false"
+nolink "false"
+
+\end_inset
+
+ ne velja.
+ Če
+\begin_inset Formula $B\subset\mathbb{Q}$
+\end_inset
+
+,
+ se lahko zgodi,
+ da
+\begin_inset Formula $\sup B\not\in\mathbb{Q}$
+\end_inset
+
+.
+ Primer:
+
+\begin_inset Formula $B\coloneqq\left\{ q\in\mathbb{Q};q^{2}\leq2\right\} $
+\end_inset
+
+.
+
+\begin_inset Formula $\sup B=\sqrt{2}\not\in\mathbb{Q}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Example*
+
+\end_layout
+
+\begin_layout Subsection
+Decimalni zapis
+\end_layout
+
+\begin_layout Definition*
+\begin_inset Formula $\forall x\in\mathbb{R}^{+}\exists!m\in\mathbb{N}\cup\left\{ 0\right\} ,d_{1},d_{2},\dots\in\left\{ 0..9\right\} $
+\end_inset
+
+,
+ ki število natančno določajo.
+ Pišemo
+\begin_inset Formula $x=m,d_{1}d_{2}\dots$
+\end_inset
+
+.
+ Natančno določitev mislimo v smislu:
+\end_layout
+
+\begin_deeper
+\begin_layout Itemize
+\begin_inset Formula $m\leq x<m+1$
+\end_inset
+
+ —
+ s tem se izognemo dvojnemu zapisu
+\begin_inset Formula $1=0,\overline{9}$
+\end_inset
+
+ in
+\begin_inset Formula $1=1,\overline{0}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Formula $[m,m+1)$
+\end_inset
+
+ razdelimo na 10 enako dolgih polodprtih intervalov
+\begin_inset Formula $I_{0},\dots,I_{9}$
+\end_inset
+
+.
+
+\begin_inset Formula $x$
+\end_inset
+
+ leži na natanko enem izmed njih,
+ indeks njega je
+\begin_inset Formula $d_{1}$
+\end_inset
+
+.
+ Nadaljujemo tako,
+ da
+\begin_inset Formula $I_{d_{1}}$
+\end_inset
+
+ razdelimo zopet na 10 delov itd.
+\end_layout
+
+\end_deeper
+\begin_layout Definition*
+Števila
+\begin_inset Formula $x\in\mathbb{R^{-}}$
+\end_inset
+
+ pišemo tako,
+ da zapišemo decimalni zapis števila
+\begin_inset Formula $-x$
+\end_inset
+
+ in predenj zapišemo
+\begin_inset Formula $-$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Definition*
+Če se decimalke v zaporedju
+\begin_inset Formula $\left(d_{n}\right)_{n\in\mathbb{N}}$
+\end_inset
+
+ ponavljajo,
+ uporabimo periodični zapis,
+ denimo
+\begin_inset Formula $5,01\overline{763}\in\mathbb{Q}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Subsection
+Kompleksna števila
+\end_layout
+
+\begin_layout Definition*
+Vpeljimo število
+\begin_inset Formula $i$
+\end_inset
+
+ z lastnostjo
+\begin_inset Formula $i^{2}=-1$
+\end_inset
+
+,
+ da je
+\begin_inset Formula $i$
+\end_inset
+
+ rešitev enačbe
+\begin_inset Formula $x^{2}+1=0$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Claim*
+\begin_inset Formula $i\not\in\mathbb{R}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Proof
+Sicer bi veljajo
+\begin_inset Formula $i^{2}\geq0$
+\end_inset
+
+,
+ kar po definiciji ne velja.
+\end_layout
+
+\begin_layout Definition*
+Kompleksna števila so
+\begin_inset Formula $\mathbb{C}\coloneqq\left\{ a+bi;a,b\in\mathbb{R}\right\} $
+\end_inset
+
+.
+
+\begin_inset Formula $bi$
+\end_inset
+
+ je še nedefinirano,
+ zato za kompleksna števila definirano seštevanje in množenje za
+\begin_inset Formula $z=a+bi$
+\end_inset
+
+ in
+\begin_inset Formula $w=c+di$
+\end_inset
+
+:
+\end_layout
+
+\begin_deeper
+\begin_layout Itemize
+\begin_inset Formula $z+w\coloneqq\left(a+c\right)+\left(b+d\right)i$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Formula $zw\coloneqq\left(a+bi\right)\left(c+di\right)=ac+adi+bic+bidi=ac+adi+bic-bd=\left(ac-bd\right)+\left(ad+bc\right)i$
+\end_inset
+
+
+\end_layout
+
+\end_deeper
+\begin_layout Definition*
+Definiramo še konjugirano vrednost
+\begin_inset Formula $z\in\mathbb{C}$
+\end_inset
+
+:
+
+\begin_inset Formula $\overline{z}\coloneqq a-bi$
+\end_inset
+
+ in označimo
+\begin_inset Formula $\left|z\right|^{2}=z\overline{z}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Claim*
+\begin_inset Formula $z\overline{z}=a^{2}+b^{2}\geq0$
+\end_inset
+
+ za
+\begin_inset Formula $z=a+bi$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Proof
+\begin_inset Formula $\left(a+bi\right)\left(a-bi\right)=a^{2}+abi-bia-bibi=a^{2}+b^{2}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Velja,
+ da je
+\begin_inset Formula $\mathbb{R}\subset\mathbb{C}$
+\end_inset
+
+ v smislu identifikacije
+\begin_inset Formula $\mathbb{R}$
+\end_inset
+
+ z množico
+\begin_inset Formula $\mathbb{C}$
+\end_inset
+
+:
+
+\begin_inset Formula $\mathbb{R}=\left\{ a+0i;a\in\mathbb{R}\right\} $
+\end_inset
+
+,
+ torej smo
+\begin_inset Formula $\mathbb{R}$
+\end_inset
+
+ razširili v
+\begin_inset Formula $\mathbb{C}$
+\end_inset
+
+,
+ kjer ima vsak polinom vedno rešitev.
+\end_layout
+
+\begin_layout Subsubsection
+Deljenje v
+\begin_inset Formula $\mathbb{C}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Za
+\begin_inset Formula $w,z\in\mathbb{C},w\not=0$
+\end_inset
+
+ iščemo
+\begin_inset Formula $x\in\mathbb{C}\ni:wx=z$
+\end_inset
+
+.
+ Ločimo dva primera:
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Formula $w\in\mathbb{R}\setminus\left\{ 0\right\} $
+\end_inset
+
+:
+ definiramo
+\begin_inset Formula $x=\frac{z}{w}\coloneqq\frac{a}{w}+\frac{b}{w}i$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Formula $w\in\mathbb{C}\setminus\left\{ 0\right\} $
+\end_inset
+
+ (splošno):
+
+\begin_inset Formula $wx=z\overset{/\cdot\overline{w}}{\Longrightarrow}w\overline{w}x=z\overline{w}\Rightarrow\left|w\right|^{2}x=z\overline{w}\Rightarrow x=\frac{z\overline{w}}{\left|w\right|^{2}}$
+\end_inset
+
+,
+ z
+\begin_inset Formula $\left|w\right|^{2}$
+\end_inset
+
+ pa znamo deliti,
+ ker je realen.
+\end_layout
+
+\begin_layout Subsubsection
+Lastnosti v
+\begin_inset Formula $\mathbb{C}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $+$
+\end_inset
+
+ in
+\begin_inset Formula $\cdot$
+\end_inset
+
+ sta komutativni,
+ asociativni,
+ distributivni,
+
+\begin_inset Formula $0$
+\end_inset
+
+ je aditivna enota,
+
+\begin_inset Formula $1$
+\end_inset
+
+ je multiplikativna.
+\end_layout
+
+\begin_layout Definition*
+Za
+\begin_inset Formula $z=a+bi$
+\end_inset
+
+ vpeljemo
+\begin_inset Formula $\Re z=a$
+\end_inset
+
+ in
+\begin_inset Formula $\Im z=b$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Remark*
+Opazimo
+\begin_inset Formula $\Re z=\frac{z+\overline{z}}{2}$
+\end_inset
+
+,
+
+\begin_inset Formula $\Im z=\frac{z-\overline{z}}{2}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Separator plain
+\end_inset
+
+
+\end_layout
+
+\begin_layout Remark*
+\begin_inset Formula $\mathbb{C}$
+\end_inset
+
+ si lahko predstavljamo kot urejene pare;
+
+\begin_inset Formula $a+bi$
+\end_inset
+
+ ustreza paru
+\begin_inset Formula $\left(a,b\right)$
+\end_inset
+
+.
+ Tako
+\begin_inset Formula $\mathbb{C}$
+\end_inset
+
+ enačimo/identificiramo z
+\begin_inset Formula $\mathbb{R}^{2}\coloneqq\left\{ \left(a,b\right);a,b\in\mathbb{R}\right\} $
+\end_inset
+
+,
+ s čimer dobimo geometrično predstavitev
+\begin_inset Formula $\mathbb{C}$
+\end_inset
+
+ kot vektorje v
+\begin_inset Formula $\mathbb{R}^{2}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Definition*
+Za
+\begin_inset Formula $z=a+bi$
+\end_inset
+
+,
+ predstavljen z vektorjem s komponentami
+\begin_inset Formula $\left(a,b\right)$
+\end_inset
+
+,
+ velja
+\begin_inset Formula $a=\left|z\right|\cos\varphi$
+\end_inset
+
+ in
+\begin_inset Formula $v=\left|z\right|\sin\varphi$
+\end_inset
+
+.
+ Kotu
+\begin_inset Formula $\varphi$
+\end_inset
+
+ pravimo argument kompleksnega števila
+\begin_inset Formula $z$
+\end_inset
+
+,
+ oznaka
+\begin_inset Formula $\arg z$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Corollary*
+\begin_inset Formula $z=$
+\end_inset
+
+
+\begin_inset Formula $\left|z\right|\left(\cos\varphi+i\sin\varphi\right)$
+\end_inset
+
+.
+ Velja
+\begin_inset Foot
+status open
+
+\begin_layout Plain Layout
+TODO DOPISATI ZAKAJ (v bistvu še jaz ne vem).
+ ne razumem.
+\end_layout
+
+\end_inset
+
+
+\begin_inset Formula $\left(\cos\varphi+i\sin\varphi\right)\left(\cos\psi+i\sin\psi\right)=\cos\left(\varphi+\psi\right)+i\sin\left(\varphi+\psi\right)$
+\end_inset
+
+,
+ zato lahko pišemo
+\begin_inset Formula $e^{i\varphi}=\cos\varphi+i\sin\varphi$
+\end_inset
+
+.
+ Množenje kompleksnh števil
+\begin_inset Formula $z=\left|z\right|e^{i\varphi}$
+\end_inset
+
+ in
+\begin_inset Formula $w=\left|w\right|e^{i\psi}$
+\end_inset
+
+ vrne število
+\begin_inset Formula $zw$
+\end_inset
+
+,
+ za katero velja:
+\end_layout
+
+\begin_deeper
+\begin_layout Itemize
+\begin_inset Formula $\left|zw\right|=\left|z\right|\left|w\right|$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Formula $\arg zw=\arg z+\arg w$
+\end_inset
+
+ (do periode
+\begin_inset Formula $2\pi$
+\end_inset
+
+ natančno)
+\end_layout
+
+\end_deeper
+\begin_layout Section
+Zaporedja
+\end_layout
+
+\begin_layout Definition*
+Funkcija
+\begin_inset Formula $a:\mathbb{N}\to\mathbb{R}$
+\end_inset
+
+ se imenuje realno zaporedje,
+ oznaka
+\begin_inset Formula $\left(a_{n}\right)_{n\in\mathbb{N}}$
+\end_inset
+
+.
+
+\begin_inset Formula $a_{n}$
+\end_inset
+
+ je funkcijska vrednost pri
+\begin_inset Formula $n$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Example*
+\begin_inset Formula $a_{n}=n$
+\end_inset
+
+:
+
+\begin_inset Formula $1,2,3,\dots$
+\end_inset
+
+;
+
+\begin_inset Formula $a_{n}=\left(-1\right)^{n}n^{2}$
+\end_inset
+
+:
+
+\begin_inset Formula $-1,4,-9,16,-25,\dots$
+\end_inset
+
+;
+
+\begin_inset Formula $a_{n}=\cos\left(\frac{\pi}{2}n\right)=0,-1,0,1,0,-1,\dots$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Zaporedje lahko podamo rekurzivno.
+ Podamo prvi člen ali nekaj prvih členov in pravilo,
+ kako iz prejšnjih členov dobiti naslednje.
+\end_layout
+
+\begin_layout Example*
+\begin_inset Formula $a_{1}=0,a_{n+1}=a_{n}+n$
+\end_inset
+
+ da zaporedje
+\begin_inset Formula $a_{n}=\frac{n\left(n+1\right)}{2}$
+\end_inset
+
+.
+
+\begin_inset Formula $a_{0}=0,a_{n+1}=\sqrt{b+a_{n}}$
+\end_inset
+
+ da zaporedje
+\begin_inset Formula $0,\sqrt{b},\sqrt{b+\sqrt{b}},\sqrt{b+\sqrt{b+\sqrt{b}}},\dots$
+\end_inset
+
+.
+ Fibbonacijevo zaporedje:
+
+\begin_inset Formula $a_{1}=a_{2}=1,a_{n+1}=a_{n}+a_{n-1}$
+\end_inset
+
+ da zaporedje
+\begin_inset Formula $1,1,2,3,5,8,\dots$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsection
+Posebni tipi zaporedij
+\end_layout
+
+\begin_layout Definition*
+Zaporedje
+\begin_inset Formula $\left(a_{n}\right)_{n\in\mathbb{N}}$
+\end_inset
+
+ je aritmetično,
+ če
+\begin_inset Formula $\exists k\in\mathbb{R}\forall n\in\mathbb{N}:a_{n+1}-a_{n}=k$
+\end_inset
+
+.
+ Tedaj
+\begin_inset Formula $a_{n+1}=a_{n}+k=a_{1}+nd$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Separator plain
+\end_inset
+
+
+\end_layout
+
+\begin_layout Definition*
+Zaporedje
+\begin_inset Formula $\left(a_{n}\right)_{n\in\mathbb{N}}$
+\end_inset
+
+ je geometrično,
+ če
+\begin_inset Formula $\exists\lambda\in\mathbb{R}\forall n\in\mathbb{N}:a_{n+1}=a_{n}\lambda$
+\end_inset
+
+.
+ Tedaj
+\begin_inset Formula $a_{n}=\lambda^{n-1}a_{1}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Separator plain
+\end_inset
+
+
+\end_layout
+
+\begin_layout Definition*
+Zaporedje
+\begin_inset Formula $\left(a_{n}\right)_{n\in\mathbb{N}}$
+\end_inset
+
+ je navzdol oz.
+ navzgor omejeno,
+ če je množica vseh členov tega taporedja navzgor oz.
+ navzdol omejena (glej
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "subsec:Omejenost-množic"
+plural "false"
+caps "false"
+noprefix "false"
+nolink "false"
+
+\end_inset
+
+).
+ Podobno z množico členov definiramo supremum,
+ infimum,
+ maksimum in infimum zaporedja.
+\end_layout
+
+\begin_layout Definition*
+Zaporedje je naraščajoče,
+ če
+\begin_inset Formula $\forall n\in\mathbb{N}:a_{n+1}\geq a_{n}$
+\end_inset
+
+,
+ padajoče,
+ če
+\begin_inset Formula $\forall n\in\mathbb{N}:a_{n+1}\leq a_{n}$
+\end_inset
+
+,
+ strogo naraščajoče,
+ če
+\begin_inset Formula $\forall n\in\mathbb{N}:a_{n+1}>a_{n}$
+\end_inset
+
+,
+ strogo padajoče podobno,
+ monotono,
+ če je naraščajoče ali padajoče in strogo monotono,
+ če je strogo naraščajoče ali strogo padajoče.
+\end_layout
+
+\begin_layout Subsection
+Limita zaporedja
+\end_layout
+
+\begin_layout Definition*
+Množica
+\begin_inset Formula $U\subseteq\mathbb{R}$
+\end_inset
+
+ je odprta,
+ če
+\begin_inset Formula $\forall u\in U\exists r>0\ni:\left(u-r,u+r\right)\subseteq U$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Separator plain
+\end_inset
+
+
+\end_layout
+
+\begin_layout Definition*
+Množica
+\begin_inset Formula $U\subseteq\mathbb{R}$
+\end_inset
+
+ je zaprta,
+ če je
+\begin_inset Formula $U^{\mathcal{C}}\coloneqq\mathbb{R}\setminus U$
+\end_inset
+
+ odprta.
+\end_layout
+
+\begin_layout Claim*
+Odprt interval je odprta množica.
+\end_layout
+
+\begin_layout Proof
+Za poljubna
+\begin_inset Formula $a,b\in\mathbb{R}$
+\end_inset
+
+,
+
+\begin_inset Formula $b>a$
+\end_inset
+
+,
+ naj bo
+\begin_inset Formula $u\in\left(a,b\right)$
+\end_inset
+
+ poljuben.
+ Ustrezen
+\begin_inset Formula $r$
+\end_inset
+
+ je
+\begin_inset Formula $\min\left\{ \left|r-a\right|,\left|r-b\right|\right\} $
+\end_inset
+
+,
+ da je
+\begin_inset Formula $\left(u-r,u+r\right)\subseteq U$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Claim*
+Zaprt interval je zaprt.
+\end_layout
+
+\begin_layout Proof
+Naj bosta
+\begin_inset Formula $a,b\in\mathbb{R}$
+\end_inset
+
+ poljubna in
+\begin_inset Formula $b>a$
+\end_inset
+
+.
+ Dokazujemo,
+ da je
+\begin_inset Formula $\left[a,b\right]$
+\end_inset
+
+ zaprt,
+ torej da je
+\begin_inset Formula $\left[a,b\right]^{\mathcal{C}}=\left(-\infty,a\right)\cup\left(b,\infty\right)$
+\end_inset
+
+ odprta množica.
+ Za poljuben
+\begin_inset Formula $u\in\left[a,b\right]^{\mathcal{C}}$
+\end_inset
+
+ velja,
+ da je bodisi
+\begin_inset Formula $\in\left(-\infty,a\right)$
+\end_inset
+
+ bodisi
+\begin_inset Formula $\left(b,\infty\right)$
+\end_inset
+
+,
+ kajti
+\begin_inset Formula $\left(-\infty,a\right)\cap\left(b,\infty\right)=\emptyset$
+\end_inset
+
+.
+ Po prejšnji trditvi v obeh primerih velja
+\begin_inset Formula $\exists r>0\ni:\left(u-r,u+r\right)\subseteq U$
+\end_inset
+
+,
+ torej je
+\begin_inset Formula $\left[a,b\right]^{\mathcal{C}}$
+\end_inset
+
+ res odprta,
+ torej je
+\begin_inset Formula $\left[a,b\right]$
+\end_inset
+
+ res zaprta.
+\end_layout
+
+\begin_layout Definition*
+Množica
+\begin_inset Formula $B$
+\end_inset
+
+ je okolica točke
+\begin_inset Formula $t\in\mathbb{R}$
+\end_inset
+
+,
+ če vsebuje kakšno odprto množico
+\begin_inset Formula $U$
+\end_inset
+
+,
+ ki vsebuje
+\begin_inset Formula $t$
+\end_inset
+
+,
+ torej
+\begin_inset Formula $t\in U^{\text{odp.}}\subseteq B\subseteq\mathbb{R}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Separator plain
+\end_inset
+
+
+\end_layout
+
+\begin_layout Definition*
+\begin_inset Formula $L\in\mathbb{R}$
+\end_inset
+
+ je limita zaporedja
+\begin_inset Formula $\left(a_{n}\right)_{n\in\mathbb{R}}$
+\end_inset
+
+
+\begin_inset Formula $\Leftrightarrow\forall\varepsilon>0\exists n_{0}\in\mathbb{N}\forall n\in\mathbb{N}:n\geq n_{0}\Rightarrow\left|a_{n}-L\right|<\varepsilon$
+\end_inset
+
+.
+ ZDB
+\begin_inset Formula $\forall V$
+\end_inset
+
+ okolica
+\begin_inset Formula $L\in\mathbb{R}\exists n_{0}\in\mathbb{N}\forall n\in\mathbb{N}:n\geq n_{0}\Rightarrow a_{n}\in V$
+\end_inset
+
+,
+ pravimo,
+ da
+\begin_inset Formula $\left(a_{n}\right)_{n\in\mathbb{N}}$
+\end_inset
+
+ konvergira k
+\begin_inset Formula $L$
+\end_inset
+
+ in pišemo
+\begin_inset Formula $L\coloneqq\lim_{n\to\infty}a_{n}$
+\end_inset
+
+ ali drugače
+\begin_inset Formula $a_{n}\underset{n\to\infty}{\longrightarrow}L$
+\end_inset
+
+.
+ Če zaporedje ima limito,
+ pravimo,
+ da je konvergentno,
+ sicer je divergentno.
+\end_layout
+
+\begin_layout Claim*
+Konvergentno zaporedje v
+\begin_inset Formula $\mathbb{R}$
+\end_inset
+
+ ima natanko eno limito.
+\end_layout
+
+\begin_layout Proof
+Naj bosta
+\begin_inset Formula $J$
+\end_inset
+
+ in
+\begin_inset Formula $L$
+\end_inset
+
+ limiti zaporedja
+\begin_inset Formula $\left(a_{n}\right)_{n\to\infty}$
+\end_inset
+
+.
+ Torej
+\begin_inset Foot
+status open
+
+\begin_layout Plain Layout
+Ko trdimo,
+ da obstaja
+\begin_inset Formula $n_{0}$
+\end_inset
+
+,
+ še ne vemo,
+ ali sta za
+\begin_inset Formula $L$
+\end_inset
+
+ in
+\begin_inset Formula $J$
+\end_inset
+
+ ta
+\begin_inset Formula $n_{0}$
+\end_inset
+
+ ista.
+ Ampak trditev še vedno velja,
+ ker lahko vzamemo večjega izmed njiju,
+ ako bi bila drugačna.
+\end_layout
+
+\end_inset
+
+ po definiciji
+\begin_inset Formula $\forall\varepsilon>0\exists n_{0}\in\mathbb{N}\forall n\in\mathbb{N}:n\geq n_{0}\Rightarrow\left|a_{n}-L\right|<\varepsilon\wedge\left|a_{n}-J\right|<\varepsilon$
+\end_inset
+
+.
+ Velja torej
+\begin_inset Formula $\forall\varepsilon>0:\left|J-L\right|<\varepsilon$
+\end_inset
+
+.
+ PDDRAA
+\begin_inset Formula $J\not=L$
+\end_inset
+
+.
+ Tedaj
+\begin_inset Formula $\left|J-L\right|\not=0$
+\end_inset
+
+,
+ naj bo
+\begin_inset Formula $\left|J-L\right|=k$
+\end_inset
+
+.
+ Tedaj
+\begin_inset Formula $\exists\varepsilon>0:\left|J-L\right|\not<\varepsilon$
+\end_inset
+
+,
+ ustrezen
+\begin_inset Formula $\varepsilon$
+\end_inset
+
+ je na primer
+\begin_inset Formula $\frac{\left|J-L\right|}{2}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Claim*
+\begin_inset CommandInset label
+LatexCommand label
+name "Konvergentno-zaporedje-v-R-je-omejeno"
+
+\end_inset
+
+Konvergentno zaporedje v
+\begin_inset Formula $\mathbb{R}$
+\end_inset
+
+ je omejeno.
+\end_layout
+
+\begin_layout Proof
+Naj bo
+\begin_inset Formula $L=\lim_{n\to\infty}a_{n}$
+\end_inset
+
+.
+ Znotraj intervala
+\begin_inset Formula $\left(L-1,L+1\right)$
+\end_inset
+
+ so vsi členi zaporedja razen končno mnogo (
+\begin_inset Formula $\left\{ a_{1},\dots,a_{n_{0}}\right\} $
+\end_inset
+
+).
+
+\begin_inset Formula $\left\{ a_{n}\right\} _{n\in\mathbb{N}}$
+\end_inset
+
+ je unija dveh omejenih množic;
+
+\begin_inset Formula $\left(L-1,L+1\right)$
+\end_inset
+
+ in
+\begin_inset Formula $\left\{ a_{1},\dots,a_{n_{0}}\right\} $
+\end_inset
+
+,
+ zato je tudi sama omejena.
+\end_layout
+
+\begin_layout Claim*
+Naj bosta
+\begin_inset Formula $\left(a_{n}\right)_{n\in\mathbb{N}}$
+\end_inset
+
+ in
+\begin_inset Formula $\left(b_{n}\right)_{n\in\mathbb{N}}$
+\end_inset
+
+ konvergentni zaporedji v
+\begin_inset Formula $\mathbb{R}$
+\end_inset
+
+.
+ Tedaj so tudi
+\begin_inset Formula $\left(a_{n}*b_{n}\right)_{n\in\mathbb{N}}$
+\end_inset
+
+ konvergentna in velja
+\begin_inset Formula $\lim_{n\to\infty}a_{n}*b_{n}=\lim_{n\to\infty}a_{n}*\lim_{n\to\infty}b_{n}$
+\end_inset
+
+ za
+\begin_inset Formula $*\in\left\{ +,-,\cdot\right\} $
+\end_inset
+
+.
+ Če je
+\begin_inset Formula $\lim_{n\to\infty}b_{n}\not=0$
+\end_inset
+
+,
+ isto velja tudi za deljenje.
+\end_layout
+
+\begin_layout Proof
+Naj bo
+\begin_inset Formula $a_{n}\to A$
+\end_inset
+
+ in
+\begin_inset Formula $b_{n}\to B$
+\end_inset
+
+ oziroma
+\begin_inset Formula $\forall\varepsilon>0\exists n_{1},n_{2}\ni:\left(n>n_{1}\Rightarrow\left|a_{n}-A\right|<\varepsilon\right)\wedge\left(n>n_{2}\Rightarrow\left|b_{n}-B\right|<\varepsilon\right)$
+\end_inset
+
+,
+ torej
+\begin_inset Formula $\forall\varepsilon>0\exists n_{0}=\max\left\{ n_{1},n_{2}\right\} \ni:n>n_{0}\Rightarrow\left|a_{n}-A\right|<\varepsilon\wedge\left|b_{n}-B\right|<\varepsilon$
+\end_inset
+
+.
+ Dokažimo za vse operacije:
+\end_layout
+
+\begin_deeper
+\begin_layout Labeling
+\labelwidthstring 00.00.0000
+\begin_inset Formula $+$
+\end_inset
+
+ Po predpostavki velja
+\begin_inset Formula $\forall\varepsilon>0\exists n_{0}\ni:n>n_{0}\Rightarrow\left|a_{n}-A\right|+\left|a_{n}-B\right|<2\varepsilon$
+\end_inset
+
+.
+ Oglejmo si sedaj
+\begin_inset Formula
+\[
+\left|\left(a_{n}+b_{n}\right)-\left(A+B\right)\right|=\left|\left(a_{n}-A\right)+\left(b_{n}-B\right)\right|\leq\left|a_{n}-A\right|+\left|b_{n}-B\right|
+\]
+
+\end_inset
+
+in uporabimo še prejšnjo trditev,
+ torej
+\begin_inset Formula $\forall2\varepsilon\exists n_{0}\ni:\left|\left(a_{n}+b_{n}\right)-\left(A+B\right)\right|\leq2\varepsilon$
+\end_inset
+
+,
+ s čimer dokažemo
+\begin_inset Formula $\left(a_{n}+b_{n}\right)\to A+B$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Labeling
+\labelwidthstring 00.00.0000
+\begin_inset Formula $-$
+\end_inset
+
+ Oglejmo si
+\begin_inset Formula
+\[
+\left|\left(a_{n}-b_{n}\right)-\left(A-B\right)\right|=\left|a_{n}-b_{n}-A+B\right|=\left|\left(a_{n}-A\right)+\left(-\left(b_{n}-B\right)\right)\right|\leq\left|a_{n}-A\right|+\left|b_{n}-B\right|
+\]
+
+\end_inset
+
+in nato kot zgoraj.
+\end_layout
+
+\begin_layout Labeling
+\labelwidthstring 00.00.0000
+\begin_inset Formula $\cdot$
+\end_inset
+
+ Oglejmo si
+\begin_inset Formula
+\[
+\left|a_{n}b_{n}-AB\right|=\left|a_{n}b_{n}-Ab_{n}+Ab_{n}-AB\right|=\left|\left(a_{n}-A\right)b_{n}+A\left(b_{n}-B\right)\right|\leq\left|a_{n}-A\right|\left|b_{n}\right|+\left|A\right|\left|b_{n}-B\right|.
+\]
+
+\end_inset
+
+Od prej vemo,
+ da sta zaporedji omejeni,
+ ker sta konvergentni,
+ zato
+\begin_inset Formula $\exists M>0\forall n\in\mathbb{N}:\left|b_{n}\right|\leq M$
+\end_inset
+
+.
+ Naj bo
+\begin_inset Formula $\varepsilon>0$
+\end_inset
+
+ poljuben
+\begin_inset Formula $n_{1},n_{2}\in\mathbb{N}$
+\end_inset
+
+ taka,
+ da
+\begin_inset Formula $n\geq n_{1}\Rightarrow\left|a_{n}-A\right|<\frac{\varepsilon}{2M}$
+\end_inset
+
+ in
+\begin_inset Formula $n\geq n_{2}\Rightarrow\left|b_{n}-B\right|<\frac{\varepsilon}{2\left|A\right|}$
+\end_inset
+
+.
+ Tedaj za
+\begin_inset Formula $n_{0}\coloneqq\max\left\{ n_{1},n_{2}\right\} $
+\end_inset
+
+ velja
+\begin_inset Formula $\left|a_{n}b_{n}-AB\right|\leq\left|a_{n}-A\right|\left|b_{n}\right|+\left|A\right|\left|b_{n}-B\right|<\frac{\varepsilon}{2M}M+\left|A\right|\frac{\varepsilon}{2\left|A\right|}=\varepsilon$
+\end_inset
+
+,
+ skratka
+\begin_inset Formula $\left|a_{n}b_{n}-AB\right|<\varepsilon$
+\end_inset
+
+,
+ s čimer dokažemo
+\begin_inset Formula $\left(a_{n}+b_{n}\right)\to A+B$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Labeling
+\labelwidthstring 00.00.0000
+\begin_inset Formula $/$
+\end_inset
+
+ Ker je
+\begin_inset Formula $B\not=0$
+\end_inset
+
+,
+
+\begin_inset Formula $\exists n_{0}\in\mathbb{N}\forall n\geq n_{0}:\left|b_{n}\right|\geq\frac{\left|B\right|}{2}>0$
+\end_inset
+
+.
+ ZDB vsi členi zaporedja razen končno mnogo so v poljubno majhni okolici
+\begin_inset Formula $\left|B\right|$
+\end_inset
+
+.
+ Če torej vzamemo točko na polovici med 0 in
+\begin_inset Formula $\left|B\right|$
+\end_inset
+
+,
+ to je
+\begin_inset Formula $\frac{\left|B\right|}{2}$
+\end_inset
+
+,
+ bo neskončno mnogo absolutnih vrednosti členov večjih od
+\begin_inset Formula $\frac{\left|B\right|}{2}$
+\end_inset
+
+.
+ Pri razumevanju pomaga številska premica.
+ Nadalje uporabimo predpostavko z
+\begin_inset Formula $\varepsilon=\frac{\left|B\right|}{2}$
+\end_inset
+
+,
+ torej je za
+\begin_inset Formula $n>n_{0}:$
+\end_inset
+
+
+\begin_inset Formula $\left|B-b_{n}\right|<\frac{\left|B\right|}{2}$
+\end_inset
+
+ in velja
+\begin_inset Formula
+\[
+\left|b_{n}\right|=\left|B-\left(B-b_{n}\right)\right|=\left|B+\left(-\left(B-b_{n}\right)\right)\right|\overset{\text{trik. neen.}}{\geq}\left|\left|B\right|-\left|B-b_{n}\right|\right|=\left|B\right|-\left|B-b_{n}\right|\overset{\text{predp.}}{>}\left|B\right|-\frac{\left|B\right|}{2}=\frac{\left|B\right|}{2},
+\]
+
+\end_inset
+
+skratka
+\begin_inset Formula $\left|b_{n}\right|>\frac{\left|B\right|}{2}$
+\end_inset
+
+.
+ Če spet izpustimo končno začetnih členov,
+ velja
+\begin_inset Formula
+\[
+\frac{a_{n}}{b_{n}}-\frac{A}{B}=\frac{a_{n}B-Ab_{n}}{b_{n}B}\overset{\text{prištejemo in odštejemo člen}}{=}\frac{\left(a_{n}-A\right)B+A\left(B-b_{n}\right)}{Bb_{n}}=\frac{1}{b_{n}}\left(a_{n}-A\right)+\frac{A/B}{b_{n}}\left(B-b_{n}\right)
+\]
+
+\end_inset
+
+sedaj uporabimo na obeh straneh absolutno vrednost:
+\begin_inset Formula
+\[
+\left|\frac{a_{n}}{b_{n}}-\frac{A}{B}\right|=\left|\frac{1}{b_{n}}\left(a_{n}-A\right)+\frac{A}{Bb_{n}}\left(B-b_{n}\right)\right|\leq\frac{1}{\left|b_{n}\right|}\left|a_{n}-A\right|+\frac{\left|A\right|}{\left|B\right|\left|b_{n}\right|}\left|B-b_{n}\right|<\frac{2}{\left|B\right|}\left|a_{n}-A\right|+\frac{2\left|A\right|}{\left|B\right|^{2}}\left|B-B_{n}\right|
+\]
+
+\end_inset
+
+skratka
+\begin_inset Formula $\left|\frac{a_{n}}{b_{n}}-\frac{A}{B}\right|<\frac{2}{\left|B\right|}\left|a_{n}-A\right|+\frac{2\left|A\right|}{\left|B\right|^{2}}\left|B-B_{n}\right|$
+\end_inset
+
+.
+ Opazimo,
+ da
+\begin_inset Formula $\frac{2}{\left|B\right|}$
+\end_inset
+
+ in
+\begin_inset Formula $\frac{2\left|A\right|}{\left|B\right|^{2}}$
+\end_inset
+
+ nista odvisna od
+\begin_inset Formula $n$
+\end_inset
+
+.
+ Sedaj vzemimo poljuben
+\begin_inset Formula $\varepsilon>0$
+\end_inset
+
+ in
+\begin_inset Formula $n_{1},n_{2}\in\mathbb{N}$
+\end_inset
+
+ takšna,
+ da velja:
+\end_layout
+
+\begin_deeper
+\begin_layout Itemize
+\begin_inset Formula $n\geq n_{1}\Rightarrow\left|a_{n}-A\right|<\frac{\varepsilon\left|B\right|}{4}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Formula $n\geq n_{2}\Rightarrow\left|b_{n}-B\right|<\frac{\varepsilon\left|B\right|^{2}}{4\left|A\right|}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Tedaj iz zgornje ocene sledi za
+\begin_inset Formula $n\geq\max\left\{ n_{0},n_{1},n_{2}\right\} $
+\end_inset
+
+:
+\begin_inset Formula
+\[
+\left|\frac{a_{n}}{b_{n}}-\frac{A}{B}\right|<\frac{\cancel{2}}{\cancel{\left|B\right|}}\cdot\frac{\varepsilon\cancel{\left|B\right|}}{\cancelto{2}{4}}+\frac{\cancel{2}\cancel{\left|A\right|}}{\cancel{\left|B\right|^{2}}}\cdot\frac{\varepsilon\cancel{\left|B\right|^{2}}}{\cancelto{2}{4}\cancel{\left|A\right|}}=\frac{\varepsilon}{2}+\frac{\varepsilon}{2}=\varepsilon,
+\]
+
+\end_inset
+
+s čimer dokažemo
+\begin_inset Formula $a_{n}/b_{n}\to A/B$
+\end_inset
+
+.
+\end_layout
+
+\end_deeper
+\end_deeper
+\begin_layout Example*
+Naj bo
+\begin_inset Formula $a>0$
+\end_inset
+
+.
+ Izračunajmo
+\begin_inset Formula
+\[
+\sqrt{a+\sqrt{a+\sqrt{a+\sqrt{\cdots}}}}\eqqcolon\alpha.
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Example*
+\begin_inset Formula $\alpha$
+\end_inset
+
+ je torej
+\begin_inset Formula $\lim_{n\to\infty}x_{n}$
+\end_inset
+
+,
+ kjer je
+\begin_inset Formula $x_{0}=0,x_{1}=\sqrt{a},x_{2}=\sqrt{a+\sqrt{a}},x_{3}=\sqrt{a+\sqrt{a+\sqrt{a}}},\dots,x_{n+1}=\sqrt{a+x_{n}}$
+\end_inset
+
+.
+ Iz zadnjega sledi
+\begin_inset Formula $x_{n+1}^{2}=a+x_{n}$
+\end_inset
+
+.
+ Če torej limita
+\begin_inset Formula $\alpha\coloneqq\lim x_{n}$
+\end_inset
+
+ obstaja,
+ mora veljati
+\begin_inset Formula $\alpha^{2}=a+\alpha$
+\end_inset
+
+ oziroma
+\begin_inset Formula $\alpha_{1,2}=\frac{1\pm\sqrt{1+4a}}{2}$
+\end_inset
+
+.
+ Opcija z minusom ni mogoča,
+ ker je zaporedje
+\begin_inset Formula $\left(x_{n}\right)_{n\in\mathbb{N}}$
+\end_inset
+
+ očitno pozitivno.
+ Če torej limita obstaja (
+\series bold
+česar še nismo dokazali
+\series default
+),
+ je enaka
+\begin_inset Formula $\frac{1+\sqrt{1+4a}}{2}$
+\end_inset
+
+,
+ za primer
+\begin_inset Formula $a=2$
+\end_inset
+
+ je torej
+\begin_inset Formula $\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{\cdots}}}}=2$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Remark*
+Lahko se zgodi,
+ da limita rekurzivno podanega zaporedja ne obstaja,
+ čeprav jo znamo izračunati,
+ če bi obstajala.
+ Na primer
+\begin_inset Formula $y_{1}\coloneqq1$
+\end_inset
+
+,
+
+\begin_inset Formula $y_{n+1}=1-2y_{n}$
+\end_inset
+
+ nam da zaporedje
+\begin_inset Formula $1,-1,3,-5,11,\dots$
+\end_inset
+
+,
+ kar očitno nima limite.
+ Če bi limita obstajala,
+ bi zanjo veljalo
+\begin_inset Formula $\beta=1-2\beta$
+\end_inset
+
+ oz.
+
+\begin_inset Formula $3\beta=1$
+\end_inset
+
+,
+
+\begin_inset Formula $\beta=\frac{1}{3}$
+\end_inset
+
+.
+ Navedimo torej nekaj zadostnih in potrebnih pogojev za konvergenco zaporedij.
+\end_layout
+
+\begin_layout Theorem*
+Naj bo
+\begin_inset Formula $\left(a_{n}\right)_{n\in\mathbb{N}}$
+\end_inset
+
+ monotono realno zaporedje.
+ Če narašča,
+ ima limito
+\begin_inset Formula $\lim_{n\to\infty}a_{n}=\sup\left\{ a_{n},n\in\mathbb{N}\right\} $
+\end_inset
+
+.
+ Če pada,
+ ima limito
+\begin_inset Formula $\lim_{n\to\infty}a_{n}=\inf\left\{ a_{n},n\in\mathbb{N}\right\} $
+\end_inset
+
+.
+ (
+\begin_inset Formula $\sup$
+\end_inset
+
+ in
+\begin_inset Formula $\inf$
+\end_inset
+
+ imata lahko tudi vrednost
+\begin_inset Formula $\infty$
+\end_inset
+
+ in
+\begin_inset Formula $-\infty$
+\end_inset
+
+ —
+ zaporedje s tako limito ni konvergentno v
+\begin_inset Formula $\mathbb{R}$
+\end_inset
+
+).
+\end_layout
+
+\begin_layout Proof
+Denimo,
+ da
+\begin_inset Formula $\left(a_{n}\right)_{n\in\mathbb{N}}$
+\end_inset
+
+ narašča.
+ Pišimo
+\begin_inset Formula $s\coloneqq\sup_{n\in\mathbb{N}}a_{n}$
+\end_inset
+
+.
+ Vzemimo poljuben
+\begin_inset Formula $\varepsilon>0$
+\end_inset
+
+.
+ Tedaj
+\begin_inset Formula $s-\varepsilon$
+\end_inset
+
+ ni zgornja meja za
+\begin_inset Formula $\left(a_{n}\right)_{n\in\mathbb{N}}$
+\end_inset
+
+,
+ zato
+\begin_inset Formula $\exists n_{0}\in\mathbb{N}\ni:s-\varepsilon<a_{n_{0}}$
+\end_inset
+
+.
+ Ker pa je
+\begin_inset Formula $\left(a_{n}\right)_{n\in\mathbb{N}}$
+\end_inset
+
+naraščajoče,
+ sledi
+\begin_inset Formula $\forall n\geq n_{0}:a_{n}\geq a_{n_{0}}>s-\varepsilon$
+\end_inset
+
+.
+ Hkrati je
+\begin_inset Formula $a_{n}\leq s$
+\end_inset
+
+,
+ saj je
+\begin_inset Formula $s$
+\end_inset
+
+ zgornja meja.
+ Torej
+\begin_inset Formula $\forall n\geq n_{0}:a_{n}\in(s-\varepsilon,s]\subset\left(s-\varepsilon,s+\varepsilon\right)$
+\end_inset
+
+,
+ s čimer dokažemo konvergenco.
+\end_layout
+
+\begin_layout Proof
+Denimo sedaj,
+ da
+\begin_inset Formula $\left(a_{n}\right)_{n\in\mathbb{N}}$
+\end_inset
+
+ pada.
+ Dokaz je povsem analogen.
+ Pišimo
+\begin_inset Formula $m\coloneqq\inf_{n\in\mathbb{N}}a_{n}$
+\end_inset
+
+.
+ Vzemimo poljuben
+\begin_inset Formula $\varepsilon>0$
+\end_inset
+
+.
+ Tedaj
+\begin_inset Formula $m+\varepsilon$
+\end_inset
+
+ ni spodnja meja za
+\begin_inset Formula $\left(a_{n}\right)_{n\in\mathbb{N}}$
+\end_inset
+
+,
+ zato
+\begin_inset Formula $\exists n_{0}\in\mathbb{N}\ni:m+\varepsilon>a_{n_{0}}$
+\end_inset
+
+.
+ Ker pa je
+\begin_inset Formula $\left(a_{n}\right)_{n\in\mathbb{N}}$
+\end_inset
+
+ padajoče,
+ sledi
+\begin_inset Formula $\forall n\geq n_{0}:a_{n}\leq a_{n_{0}}<m+\varepsilon$
+\end_inset
+
+.
+ Hkrati je
+\begin_inset Formula $a_{n}\geq m$
+\end_inset
+
+,
+ saj je
+\begin_inset Formula $m$
+\end_inset
+
+ spodnja meja.
+ Torej
+\begin_inset Formula $\forall n\geq n_{0}:a_{n}\in[m,m+\varepsilon)\subset\left(m-\varepsilon,m+\varepsilon\right)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Corollary*
+Za monotono zaporedje velja,
+ da je v
+\begin_inset Formula $\mathbb{R}$
+\end_inset
+
+ konvergentno natanko tedaj,
+ ko je omejeno.
+\end_layout
+
+\begin_layout Example*
+Naj bo,
+ kot prej,
+
+\begin_inset Formula $a>0$
+\end_inset
+
+ in
+\begin_inset Formula $x_{0}=0,x_{n+1}=\sqrt{a+x_{n}}$
+\end_inset
+
+.
+ Dokažimo,
+ da je
+\begin_inset Formula $\left(x_{n}\right)_{n}$
+\end_inset
+
+ konvergentno.
+ Dovolj je pokazati,
+ da je naraščajoče in navzgor omejeno.
+\end_layout
+
+\begin_deeper
+\begin_layout Itemize
+Naraščanje z indukcijo:
+ Baza:
+
+\begin_inset Formula $0=x_{0}>x_{1}=\sqrt{a}$
+\end_inset
+
+.
+ Dokažimo
+\begin_inset Formula $x_{n+1}-x_{n}>0$
+\end_inset
+
+.
+\begin_inset Formula
+\[
+\left(x_{n+1}-x_{n}\right)\left(x_{n+1}+x_{n}\right)=x_{n+1}^{2}-x_{n}^{2}=\left(a+x_{n}\right)-\left(a+x_{n-1}\right)=x_{n}-x_{n-1}
+\]
+
+\end_inset
+
+Ker je zaporedje pozitivno,
+ je
+\begin_inset Formula $x_{n+1}+x_{n}>0$
+\end_inset
+
+.
+ Desna stran je po I.
+ P.
+ pozitivna,
+ torej tudi
+\begin_inset Formula $x_{n+1}-x_{n}>0$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+Omejenost:
+ Če je zaporedje res omejeno,
+ je po zgornjem tudi konvergentno in je
+\begin_inset Formula $\sup_{n\in\mathbb{N}}x_{n}=\lim_{n\to\infty}x_{n}=\frac{1+\sqrt{1+4a}}{2}\leq\frac{1+\sqrt{1+4a+4a^{2}}}{2}=\frac{1+\sqrt{\left(2a+1\right)^{2}}}{}=1+a$
+\end_inset
+
+.
+ Uganili smo neko zgornjo mejo.
+ Domnevamo,
+ da
+\begin_inset Formula $\forall n\in\mathbb{N}:x_{n}\leq1+a$
+\end_inset
+
+.
+ Dokažimo to z indukcijo:
+ Baza:
+
+\begin_inset Formula $0=x_{0}<1+a$
+\end_inset
+
+.
+ Po I.
+ P.
+
+\begin_inset Formula $x_{n}>1+a$
+\end_inset
+
+.
+ Korak:
+\begin_inset Formula
+\[
+x_{n+1}=\sqrt{x_{n}+a}\leq\sqrt{1+a+a}=\sqrt{1+2a}<\sqrt{1+2a+2a^{2}}=1+a
+\]
+
+\end_inset
+
+
+\end_layout
+
+\end_deeper
+\begin_layout Example*
+S tem smo dokazali,
+ da
+\begin_inset Formula $\lim_{n\to\infty}x_{n}=\frac{1+\sqrt{1+4a}}{2}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Example*
+To lahko dokažemo tudi na alternativen način.
+ Vidimo,
+ da je edini kandidat za limito,
+ če obstaja
+\begin_inset Formula $L=\frac{1+\sqrt{1+4a}}{2}$
+\end_inset
+
+ in da torej velja
+\begin_inset Formula $L^{2}=a+L$
+\end_inset
+
+.
+ Preverimo,
+ da je
+\begin_inset Formula $L$
+\end_inset
+
+ res limita:
+
+\begin_inset Formula
+\[
+x_{n+1}-L=\sqrt{a+x_{n}}-L=\frac{\left(\sqrt{a+x_{n}}-L\right)\left(\sqrt{a+x_{n}}+L\right)}{\sqrt{a+x_{n}}+L}=\frac{\left(a+x_{n}\right)-L^{2}}{\sqrt{a+x_{n}}+L}=\frac{\left(a+x_{n}\right)-\left(a+L\right)}{\sqrt{a+x_{n}}+L}=\frac{x_{n}-L}{\sqrt{a+x_{n}}+L}.
+\]
+
+\end_inset
+
+Vpeljimo sedaj
+\begin_inset Formula $y_{n}\coloneqq x_{n}-L$
+\end_inset
+
+.
+ Sledi
+\begin_inset Formula $\left|y_{n+1}\right|\leq\frac{\left|y_{n}\right|}{\sqrt{a+x_{n}}+L}\leq\frac{\left|y_{n}\right|}{L}$
+\end_inset
+
+.
+ Ker je
+\begin_inset Formula $\left|y_{0}\right|=L$
+\end_inset
+
+,
+ dobimo
+\begin_inset Foot
+status open
+
+\begin_layout Plain Layout
+Za razumevanje si oglej nekaj členov rekurzivnega zaporedje
+\begin_inset Formula $y_{0}=L,y_{n}=\frac{\left|y_{n+1}\right|}{L}$
+\end_inset
+
+.
+ Začnemo z 1 in nato vsakič delimo z
+\begin_inset Formula $L$
+\end_inset
+
+.
+\end_layout
+
+\end_inset
+
+ oceno
+\begin_inset Formula $\left|y_{n}\right|\leq\frac{1}{L^{n-1}}$
+\end_inset
+
+ oziroma
+\begin_inset Formula $\left|x_{n}-L\right|\leq\frac{1}{L^{n-1}}$
+\end_inset
+
+.
+ Ker iz definicije
+\begin_inset Formula $L$
+\end_inset
+
+ sledi
+\begin_inset Formula $L>1$
+\end_inset
+
+,
+ je
+\begin_inset Formula $L^{n}\to\infty$
+\end_inset
+
+ za
+\begin_inset Formula $n\to\infty$
+\end_inset
+
+,
+ torej smo dokazali,
+ da
+\begin_inset Formula $\left|x_{n}-L\right|$
+\end_inset
+
+ eksponentno pada proti 0 za
+\begin_inset Formula $n\to\infty$
+\end_inset
+
+.
+ Eksponentno padanje
+\begin_inset Formula $\left|x_{n}-L\right|$
+\end_inset
+
+ proti 0 je dovolj,
+ da rečemo,
+ da zaporedje konvergira k
+\begin_inset Formula $L$
+\end_inset
+
+
+\begin_inset Foot
+status open
+
+\begin_layout Plain Layout
+a res,
+ vprašaj koga.
+ ne razumem.
+ zakaj.
+ TODO.
+\end_layout
+
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Claim*
+\begin_inset Formula $\lim_{n\to\infty}\sin n$
+\end_inset
+
+ in
+\begin_inset Formula $\lim_{n\to\infty}\cos n$
+\end_inset
+
+ ne obstajata.
+\end_layout
+
+\begin_layout Proof
+Pišimo
+\begin_inset Formula $a_{n}=\sin n$
+\end_inset
+
+ in
+\begin_inset Formula $b_{n}=\cos n$
+\end_inset
+
+.
+ Iz adicijskih izrekov dobimo
+\begin_inset Formula $a_{n+1}=\sin\left(n+1\right)=\sin n\cos1+\cos n\sin1=a_{n}\cos1+b_{n}\sin1$
+\end_inset
+
+.
+ Torej
+\begin_inset Formula $b_{n}=\frac{a_{n+1}-a_{n}\cos1}{\sin1}$
+\end_inset
+
+.
+ Torej če
+\begin_inset Formula $\exists a\coloneqq\lim_{n\to\infty}a_{n},a\in\mathbb{R}$
+\end_inset
+
+,
+ potem tudi
+\begin_inset Formula $\exists b\coloneqq\lim_{n\to\infty}b_{n},b\in\mathbb{R}$
+\end_inset
+
+.
+ Podobno iz adicijske formule za
+\begin_inset Formula $\cos\left(n+1\right)$
+\end_inset
+
+ sledi
+\begin_inset Formula $a_{n}=\frac{b_{n}\cos1-b_{n+1}}{\sin1}$
+\end_inset
+
+,
+ torej če
+\begin_inset Formula $\exists b$
+\end_inset
+
+,
+ potem tudi
+\begin_inset Formula $\exists a$
+\end_inset
+
+.
+ Iz obojega sledi,
+ da
+\begin_inset Formula $\exists a\Leftrightarrow\exists b$
+\end_inset
+
+.
+ Posledično,
+ če
+\begin_inset Formula $a$
+\end_inset
+
+ in
+\begin_inset Formula $b$
+\end_inset
+
+ obstajata,
+ iz zgornjih obrazcev za
+\begin_inset Formula $a_{n}$
+\end_inset
+
+ in
+\begin_inset Formula $b_{n}$
+\end_inset
+
+ sledi,
+ da za
+\begin_inset Formula
+\[
+\lambda=\frac{1-\cos1}{\sin1}\in\left(0,1\right)
+\]
+
+\end_inset
+
+velja
+\begin_inset Formula $b=\lambda a$
+\end_inset
+
+ in
+\begin_inset Formula $a=-\lambda b$
+\end_inset
+
+ in zato
+\begin_inset Formula $b=\lambda\left(-\lambda b\right)$
+\end_inset
+
+ oziroma
+\begin_inset Formula $1=-\lambda^{2}$
+\end_inset
+
+,
+ torej
+\begin_inset Formula $-1=\lambda^{2}$
+\end_inset
+
+,
+ torej
+\begin_inset Formula $\lambda=i$
+\end_inset
+
+,
+ kar je v protislovju z
+\begin_inset Formula $\lambda\in\left(0,1\right)$
+\end_inset
+
+.
+ Podobno za
+\begin_inset Formula $a=-\lambda\left(\lambda a\right)$
+\end_inset
+
+ oziroma
+\begin_inset Formula $1=-\lambda^{2}$
+\end_inset
+
+,
+ kar je zopet
+\begin_inset Formula $\rightarrow\!\leftarrow$
+\end_inset
+
+.
+ Edina druga opcija je,
+ da je
+\begin_inset Formula $a=b=0$
+\end_inset
+
+.
+ Hkrati pa vemo,
+ da
+\begin_inset Formula $a_{n}^{2}+b_{n}^{2}=1$
+\end_inset
+
+,
+ zato
+\begin_inset Formula $a+b=1$
+\end_inset
+
+,
+ kar ni mogoče za ničelna
+\begin_inset Formula $a$
+\end_inset
+
+ in
+\begin_inset Formula $b$
+\end_inset
+
+.
+ Torej
+\begin_inset Formula $a$
+\end_inset
+
+ in
+\begin_inset Formula $b$
+\end_inset
+
+ ne obstajata.
+\end_layout
+
+\begin_layout Subsection
+Eulerjevo število
+\end_layout
+
+\begin_layout Theorem*
+Bernoullijeva neenakost.
+
+\begin_inset Formula $\forall\alpha\leq1,n\in\mathbb{N}$
+\end_inset
+
+ velja
+\begin_inset Formula $\left(1-\alpha\right)^{n}\geq1-n\alpha$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Proof
+Z indukcijo na
+\begin_inset Formula $n$
+\end_inset
+
+ ob fiksnem
+\begin_inset Formula $\alpha$
+\end_inset
+
+.
+\end_layout
+
+\begin_deeper
+\begin_layout Itemize
+Baza:
+
+\begin_inset Formula $n=1$
+\end_inset
+
+:
+
+\begin_inset Formula $\left(1-\alpha\right)^{1}=1-1\alpha$
+\end_inset
+
+.
+ Velja celo enakost.
+\end_layout
+
+\begin_layout Itemize
+I.
+ P.:
+ Velja
+\begin_inset Formula $\left(1-\alpha\right)^{n}\geq1-n\alpha$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Itemize
+Korak:
+
+\begin_inset Formula $\left(1-\alpha\right)^{n+1}=\left(1-\alpha\right)\left(1-\alpha\right)^{n}\geq\left(1-\alpha\right)$
+\end_inset
+
+
+\begin_inset Formula $\left(1-n\alpha\right)=1-n\alpha-\alpha-n\alpha^{2}=1-\left(n+1\right)\alpha-n\alpha^{2}\geq1-\left(n+1\right)\alpha$
+\end_inset
+
+,
+ torej ocena velja tudi za
+\begin_inset Formula $n+1$
+\end_inset
+
+.
+\end_layout
+
+\end_deeper
+\begin_layout Definition*
+Vpeljimo oznaki:
+\end_layout
+
+\begin_deeper
+\begin_layout Itemize
+Za
+\begin_inset Formula $n\in\mathbb{N}$
+\end_inset
+
+ označimo
+\begin_inset Formula $n!=1\cdot2\cdot3\cdot\cdots\cdot n$
+\end_inset
+
+ (pravimo
+\begin_inset Formula $n-$
+\end_inset
+
+faktoriala oziroma
+\begin_inset Formula $n-$
+\end_inset
+
+fakulteta).
+ Ker velja
+\begin_inset Formula $n!=n\cdot\left(n-1\right)!$
+\end_inset
+
+ za
+\begin_inset Formula $n\geq2$
+\end_inset
+
+,
+ je smiselno definirati še
+\begin_inset Formula $0!=1$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+Za
+\begin_inset Formula $n,k\in\mathbb{N}$
+\end_inset
+
+ označimo še binomski simbol:
+
+\begin_inset Formula $\binom{n}{k}\coloneqq\frac{n!}{k!\left(n-k\right)!}$
+\end_inset
+
+ (pravimo
+\begin_inset Formula $n$
+\end_inset
+
+ nad
+\begin_inset Formula $k$
+\end_inset
+
+).
+\end_layout
+
+\begin_layout Itemize
+Če je
+\begin_inset Formula $\left(a_{k}\right)_{k}$
+\end_inset
+
+ neko zaporedje (lahko tudi končno),
+ lahko pišemo
+\begin_inset Formula $\sum_{k=1}^{n}a_{k}\coloneqq a_{1}+a_{2}+\cdots+a_{n}$
+\end_inset
+
+ (pravimo summa) in
+\begin_inset Formula $\prod_{k=1}^{n}a_{k}\coloneqq a_{1}\cdot\cdots\cdot a_{n}$
+\end_inset
+
+ (pravimo produkt).
+\end_layout
+
+\end_deeper
+\begin_layout Example*
+\begin_inset Formula $\sum_{k=1}^{n}\frac{1}{k}=\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}$
+\end_inset
+
+ in
+\begin_inset Formula $\prod_{k=1}^{n}k=1\cdot2\cdot3\cdot\cdots\cdot n=n!$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Theorem*
+Binomska formula.
+
+\begin_inset Formula $\forall a,b\in\mathbb{R},n\in\mathbb{N}:\left(a+b\right)^{n}=\sum_{k=0}^{n}\binom{n}{k}a^{k}b^{n-k}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Proof
+Indukcija po
+\begin_inset Formula $n$
+\end_inset
+
+.
+\end_layout
+
+\begin_deeper
+\begin_layout Itemize
+Baza
+\begin_inset Formula $n=1$
+\end_inset
+
+:
+
+\begin_inset Formula $\sum_{k=0}^{1}\binom{n}{k}a^{k}b^{n-k}=\binom{1}{0}a^{0}b^{1-0}+\binom{1}{1}a^{1}b^{1-1}=a+b=\left(a+b\right)^{1}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Itemize
+I.
+ P.
+
+\begin_inset Formula $\sum_{k=0}^{n}\binom{n}{k}a^{k}b^{n-k}=\left(a+b\right)^{n}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Itemize
+Korak:
+
+\begin_inset Formula
+\[
+\left(a+b\right)^{n+1}=\left(a+b\right)\left(a+b\right)^{n}=\left(a+b\right)\sum_{k=0}^{n}\binom{n}{k}a^{k}b^{n-k}=\sum_{k=0}^{n}\binom{n}{k}a^{k+1}b^{n-k}+\sum_{k=0}^{n}\binom{n}{k}a^{k}b^{n-k+1}=
+\]
+
+\end_inset
+
+sedaj naj bo
+\begin_inset Formula $m=k+1$
+\end_inset
+
+ v levem členu:
+\begin_inset Formula
+\[
+=\sum_{m=1}^{n+1}\binom{n}{m-1}a^{m}b^{n-\left(m-1\right)}+\sum_{k=0}^{n}\binom{n}{k}a^{k}b^{n-k+1}=a^{n+1}+\sum_{k=1}^{n}\left[\binom{n}{k-1}+\binom{n}{k}\right]a^{k}b^{n-k+1}+b^{n+1}=
+\]
+
+\end_inset
+
+Sedaj obravnavajmo le izraz v oglatih oklepajih:
+\begin_inset Formula
+\[
+\binom{n}{k-1}+\binom{n}{k}=\frac{n!}{\left(k-1\right)!\left(n-k+1\right)!}+\frac{n!}{k!\left(n-k\right)!}=\frac{kn!}{k!\left(n-k+1\right)!}+\frac{n!\left(n-k+1\right)}{k!\left(n-k+1\right)!}=\frac{kn!+n!\left(n-k+1\right)}{k!\left(n-k+1\right)!}=
+\]
+
+\end_inset
+
+
+\begin_inset Formula
+\[
+=\frac{n!\left(\cancel{k+}n\cancel{-k}+1\right)}{k!\left(n+1-k\right)!}=\frac{n!\left(n+1\right)}{k!\left(n+1-k\right)!}=\frac{\left(n+1\right)!}{k!\left(n+1-k\right)}=\binom{n+1}{k}
+\]
+
+\end_inset
+
+in skratka dobimo
+\begin_inset Formula $\binom{n}{k-1}+\binom{n}{k}=\binom{n+1}{k}$
+\end_inset
+
+.
+ Vstavimo to zopet v naš zgornji račun:
+\begin_inset Formula
+\[
+\cdots=a^{n+1}+\sum_{k=1}^{n}\left[\binom{n}{k-1}+\binom{n}{k}\right]a^{k}b^{n-k+1}+b^{n+1}=a^{n+1}+\sum_{k=1}^{n}\binom{n+1}{k}a^{k}b^{n-k+1}+b^{n+1}=
+\]
+
+\end_inset
+
+
+\begin_inset Formula
+\[
+=a^{n+1}+\sum_{k=1}^{n}\binom{n+1}{k}a^{k}b^{n-k+1}+b^{n+1}=a^{n+1}+\sum_{k=0}^{n}\binom{n+1}{k}a^{k}b^{n-k+1}=\sum_{k=0}^{n+1}\binom{n+1}{k}a^{k}b^{n-k+1}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\end_deeper
+\begin_layout Theorem*
+Bernoulli.
+ Zaporedje
+\begin_inset Formula $a_{n}\coloneqq\left(1+\frac{1}{n}\right)^{n}$
+\end_inset
+
+ je konvergentno.
+\end_layout
+
+\begin_layout Proof
+Dokazali bomo,
+ da je naraščajoče in omejeno.
+\end_layout
+
+\begin_deeper
+\begin_layout Itemize
+Naraščanje:
+ Dokazujemo,
+ da za
+\begin_inset Formula $n\geq2$
+\end_inset
+
+ velja
+\begin_inset Formula $a_{n}\geq a_{n-1}$
+\end_inset
+
+ oziroma
+\begin_inset Formula
+\[
+\left(1+\frac{1}{n}\right)^{n}\overset{?}{\geq}\left(1+\frac{1}{n-1}\right)^{n-1}=\left(\frac{n-1}{n-1}+\frac{1}{n-1}\right)^{n-1}=\left(\frac{n}{n-1}\right)^{n-1}=\left(\frac{n-1}{n}\right)^{1-n}=\left(1-\frac{1}{n}\right)^{1-n}
+\]
+
+\end_inset
+
+
+\begin_inset Formula
+\[
+\left(1+\frac{1}{n}\right)^{n}\overset{?}{\geq}\left(1-\frac{1}{n}\right)^{1-n}
+\]
+
+\end_inset
+
+
+\begin_inset Formula
+\[
+\left(1+\frac{1}{n^{2}}\right)^{n}=\left(\left(1+\frac{1}{n}\right)\left(1-\frac{1}{n}\right)\right)^{n}=\left(1+\frac{1}{n}\right)^{n}\left(1-\frac{1}{n}\right)^{n}\overset{?}{\geq}1-\frac{1}{n}
+\]
+
+\end_inset
+
+
+\begin_inset Formula
+\[
+\left(1+\frac{1}{n^{2}}\right)^{n}\overset{?}{\geq}1-\frac{1}{n},
+\]
+
+\end_inset
+
+kar je poseben primer Bernoullijeve neenakosti za
+\begin_inset Formula $\alpha=\frac{1}{n^{2}}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+Omejenost:
+ Po binomski formuli je
+\begin_inset Formula
+\[
+a_{n}=\left(1+\frac{1}{n}\right)^{n}=\sum_{k=0}^{n}\binom{n}{k}\left(\frac{1}{n}\right)^{k}=\sum_{k=0}^{n}\frac{n!}{k!\left(n-k\right)!n^{k}}=1+1+\sum_{k=2}^{n}\frac{n!}{k!\left(n-k\right)!n^{k}}=2+\sum_{k=2}^{n}\frac{1}{k!}\cdot\frac{n\left(n-1\right)\left(n-2\right)\cdots\left(n-k+1\right)}{n^{k}}=
+\]
+
+\end_inset
+
+
+\begin_inset Formula
+\[
+=2+\sum_{k=2}^{n}\frac{1}{k!}\cdot\frac{n}{n}\cdot\frac{n-1}{n}\cdot\frac{n-2}{n}\cdot\cdots\cdot\frac{n-k+1}{n}=2+\sum_{k=2}^{n}\frac{1}{k!}\cdot\cancel{\left(1-0\right)}\cdot\left(1-\frac{1}{n}\right)\cdot\left(1-\frac{2}{n}\right)\cdot\cdots\cdot\left(1-\frac{k-1}{n}\right)=
+\]
+
+\end_inset
+
+
+\begin_inset Formula
+\[
+=2+\sum_{k=2}^{n}\frac{1}{k!}\prod_{j=1}^{k-1}\left(1-\frac{j}{n}\right)<2+\sum_{k=2}^{n}\frac{1}{k!}<2+\sum_{k=2}^{n}\frac{1}{2^{k-1}}=\cdots
+\]
+
+\end_inset
+
+Opomnimo,
+ da je
+\begin_inset Formula $1-\frac{j}{n}<1$
+\end_inset
+
+,
+ zato
+\begin_inset Formula $\prod_{j=1}^{k-1}\left(1-\frac{j}{n}\right)<1$
+\end_inset
+
+ (prvi neenačaj) ter
+\begin_inset Formula $k!=1\cdot2\cdot3\cdot\cdots\cdot k\geq1\cdot2\cdot2\cdot\cdots\cdot2=2^{k-1}$
+\end_inset
+
+ (drugi).
+ Sedaj si z indukcijo dokažimo
+\begin_inset Formula $\sum_{k=2}^{n}\frac{1}{2^{k-1}}=1-\frac{1}{2^{n-1}}$
+\end_inset
+
+:
+\end_layout
+
+\begin_deeper
+\begin_layout Itemize
+Baza:
+
+\begin_inset Formula $n=2$
+\end_inset
+
+:
+
+\begin_inset Formula $\frac{1}{2^{2-1}}=1-\frac{1}{2^{2-1}}=1-\frac{1}{2}=\frac{1}{2}$
+\end_inset
+
+.
+ Velja!
+\end_layout
+
+\begin_layout Itemize
+I.
+ P.:
+
+\begin_inset Formula $\sum_{k=2}^{n}\frac{1}{2^{k-1}}=1-\frac{1}{2^{n-1}}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Itemize
+Korak:
+
+\begin_inset Formula $\sum_{k=2}^{n+1}\frac{1}{2^{k-1}}=1-\frac{1}{2^{n-1}}+\frac{1}{2^{n+1-1}}=1-2\cdot2^{-n}+2^{-n}=1+2^{-n}\left(1-2\right)=1+2^{-n}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+In nadaljujmo z računanjem:
+\begin_inset Formula
+\[
+\cdots=2+1-\frac{1}{2^{n-1}}=3-\frac{1}{2^{n-1}},
+\]
+
+\end_inset
+
+s čimer dobimo zgornjo mejo
+\begin_inset Formula $\forall n\in\mathbb{N}:a_{n}<3$
+\end_inset
+
+.
+ Ker je očitno
+\begin_inset Formula $\forall n\in\mathbb{N}:a_{n}>0$
+\end_inset
+
+,
+ je torej zaporedje omejeno in ker je tudi monotono po prejšnjem izreku konvergira.
+\end_layout
+
+\end_deeper
+\end_deeper
+\begin_layout Definition*
+Označimo število
+\begin_inset Formula $e\coloneqq\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^{n}$
+\end_inset
+
+ in ga imenujemo Eulerjevo število.
+ Velja
+\begin_inset Formula $e\approx2,71828$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Remark*
+V dokazu vidimo moč izreka
+\begin_inset Quotes gld
+\end_inset
+
+omejenost in monotonost
+\begin_inset Formula $\Rightarrow$
+\end_inset
+
+ konvergenca
+\begin_inset Quotes grd
+\end_inset
+
+,
+ saj nam omogoča dokazati konvergentnost zaporedja brez kandidata za limito.
+ Jasno je,
+ da ne bi mogli vnaprej uganiti,
+ da je limita ravno
+\shape italic
+transcendentno število
+\shape default
+
+\begin_inset Formula $e$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Definition*
+Podzaporedje zaporedja
+\begin_inset Formula $\left(a_{n}\right)_{n\in\mathbb{N}}$
+\end_inset
+
+ je poljubno zaporedje oblike
+\begin_inset Formula $\left(a_{\varphi\left(n\right)}\right)_{n\in\mathbb{N}}$
+\end_inset
+
+,
+ kjer je
+\begin_inset Formula $\varphi:\mathbb{N}\to\mathbb{N}$
+\end_inset
+
+ strogo naraščajoča funkcija.
+ Definicijsko območje
+\begin_inset Formula $\varphi$
+\end_inset
+
+ mora vsebovati števno neskončno elementov.
+\end_layout
+
+\begin_layout Theorem*
+Če je
+\begin_inset Formula $L=\lim_{n\to\infty}a_{n}$
+\end_inset
+
+,
+ tedaj je
+\begin_inset Formula $L$
+\end_inset
+
+ tudi limita vsakega podzaporedja.
+\end_layout
+
+\begin_layout Proof
+Po predpostavki velja
+\begin_inset Formula $\forall\varepsilon>0\exists n_{0}\in\mathbb{N}\forall n\in\mathbb{N}:n\geq n_{0}:\left|a_{n}-L\right|<\varepsilon$
+\end_inset
+
+.
+ Vzemimo poljuben
+\begin_inset Formula $\varepsilon>0$
+\end_inset
+
+.
+ Po predpostavki obstaja
+\begin_inset Formula $n_{0}\in\mathbb{N}$
+\end_inset
+
+,
+ da bodo vsi členi zaporedja po
+\begin_inset Formula $n_{0}-$
+\end_inset
+
+tem v
+\begin_inset Formula $\left(L-\varepsilon,L+\varepsilon\right)$
+\end_inset
+
+.
+ Iz definicijskega območja
+\begin_inset Formula $\varphi$
+\end_inset
+
+ vzemimo poljuben element
+\begin_inset Formula $n_{1}$
+\end_inset
+
+,
+ da velja
+\begin_inset Formula $n_{1}\geq n_{0}$
+\end_inset
+
+.
+ Gotovo obstaja,
+ ker je definicijsko območje števno neskončne moči in s pogojem
+\begin_inset Formula $n_{1}\geq n_{0}$
+\end_inset
+
+ onemogočimo izbiro le končno mnogo elementov.
+
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+Če slednji ne obstaja,
+ je v
+\begin_inset Formula $D_{\varphi}$
+\end_inset
+
+ končno mnogo elementov,
+ tedaj vzamemo
+\begin_inset Formula $n_{1}\coloneqq\max D_{\varphi}+1$
+\end_inset
+
+ in je pogoj za limito izpolnjen na prazno.
+ Sicer pa v
+\end_layout
+
+\end_inset
+
+Velja
+\begin_inset Formula $\forall n\in\mathbb{N}:n>n_{1}\Rightarrow\left|a_{\varphi n}-L\right|<\varepsilon$
+\end_inset
+
+,
+ ker je
+\begin_inset Formula $\varphi$
+\end_inset
+
+ strogo naraščajoča in izbiramo le elemente podzaporedja,
+ ki so v izvornem zaporedju za
+\begin_inset Formula $n_{0}-$
+\end_inset
+
+tim členom in zato v
+\begin_inset Formula $\left(L-\varepsilon,L+\varepsilon\right)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Example*
+\begin_inset Formula $\lim_{n\to\infty}\frac{1}{2n+3}=\lim_{n\to\infty}\frac{1}{n}=0$
+\end_inset
+
+ za zaporedje
+\begin_inset Formula $a_{n}=\frac{1}{n}$
+\end_inset
+
+ in podzaporedje
+\begin_inset Formula $a_{\varphi n}$
+\end_inset
+
+,
+ kjer je
+\begin_inset Formula $\varphi\left(n\right)=2n+3$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Theorem*
+Karakterizacija limite s podzaporedji.
+ Naj bo
+\begin_inset Formula $\left(a_{n}\right)_{n\in\mathbb{N}}$
+\end_inset
+
+ realno zaporedje in
+\begin_inset Formula $L\in\mathbb{R}$
+\end_inset
+
+.
+ Tedaj
+\begin_inset Formula $L=\lim_{n\to\infty}a_{n}\Leftrightarrow$
+\end_inset
+
+ za vsako podzaporedje
+\begin_inset Formula $\left(a_{n_{k}}\right)_{k\in\mathbb{N}}$
+\end_inset
+
+ zaporedja
+\begin_inset Formula $\left(a_{n}\right)_{n}$
+\end_inset
+
+ obstaja njegovo podzaporedje
+\begin_inset Formula $\left(a_{n_{k_{l}}}\right)_{l\in\mathbb{N}}$
+\end_inset
+
+,
+ ki konvergira k
+\begin_inset Formula $L$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Proof
+Dokazujemo ekvivalenco:
+\end_layout
+
+\begin_deeper
+\begin_layout Labeling
+\labelwidthstring 00.00.0000
+\begin_inset Formula $\left(\Rightarrow\right)$
+\end_inset
+
+ Dokazano poprej.
+ Limita se pri prehodu na podzaporedje ohranja.
+\end_layout
+
+\begin_layout Labeling
+\labelwidthstring 00.00.0000
+\begin_inset Formula $\left(\Leftarrow\right)$
+\end_inset
+
+ PDDRAA
+\begin_inset Formula $a_{n}\not\to L$
+\end_inset
+
+.
+ Tedaj
+\begin_inset Formula $\exists\varepsilon>0$
+\end_inset
+
+ in podzaporedje
+\begin_inset Formula $\left(a_{n_{k}}\right)_{k\in\mathbb{N}}\ni:\forall k\in\mathbb{N}:\left|a_{n_{k}}-K\right|>\varepsilon$
+\end_inset
+
+ (*)
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+tu je na
+\begin_inset Quotes gld
+\end_inset
+
+Zaporedja 2
+\begin_inset Quotes grd
+\end_inset
+
+ napaka,
+ neenačaj obrne v drugo smer
+\end_layout
+
+\end_inset
+
+.
+ Po predpostavki sedaj
+\begin_inset Formula $\exists\left(a_{n_{k_{l}}}\right)_{l\in\mathbb{N}}\ni:\lim_{l\to\infty}a_{n_{k_{l}}}=L$
+\end_inset
+
+.
+ To pa je v protislovju z (*),
+ torej je začetna predpostavka
+\begin_inset Formula $a_{n}\not\to L$
+\end_inset
+
+ napačna,
+ torej
+\begin_inset Formula $a_{n}\to L$
+\end_inset
+
+.
+\end_layout
+
+\end_deeper
+\begin_layout Subsection
+Stekališča
+\end_layout
+
+\begin_layout Definition*
+Točka
+\begin_inset Formula $s\in\mathbb{R}$
+\end_inset
+
+ je stekališče zaporedje
+\begin_inset Formula $\left(a_{n}\right)_{n\in\mathbb{N}}\subset\mathbb{R}$
+\end_inset
+
+,
+ če v vsaki okolici te točke leži neskončno členov zaporedja.
+\end_layout
+
+\begin_layout Remark*
+Pri limiti zahtevamo več;
+ da izven vsake okolice limite leži le končno mnogo členov.
+\end_layout
+
+\begin_layout Example*
+Primeri stekališč.
+\end_layout
+
+\begin_deeper
+\begin_layout Enumerate
+\begin_inset Formula $L=\lim_{n\to\infty}a_{n}\Rightarrow L$
+\end_inset
+
+ je stekališče za
+\begin_inset Formula $a_{n}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $0,1,0,1,\dots$
+\end_inset
+
+ stekališči sta
+\begin_inset Formula $\left\{ 0,1\right\} $
+\end_inset
+
+ in zaporedje nima limite (ni konvergentno)
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $1,1,2,1,2,3,1,2,3,4,\dots$
+\end_inset
+
+ ima neskončno stekališč,
+
+\begin_inset Formula $\mathbb{N}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $b_{n}=n-$
+\end_inset
+
+to racionalno število
+\begin_inset Foot
+status open
+
+\begin_layout Plain Layout
+Racionalnih števil je števno mnogo,
+ zato jih lahko linearno uredimo in oštevilčimo
+\end_layout
+
+\end_inset
+
+ ima neskončno stekališč,
+
+\begin_inset Formula $\mathbb{R}$
+\end_inset
+
+.
+\end_layout
+
+\end_deeper
+\begin_layout Remark*
+Limita je stakališče,
+ stekališče pa ni nujno limita.
+ Poleg tega,
+ če se spomnimo,
+ velja,
+ da vsota konvergentnih zaporedij konvergira k vsoti njunih limit,
+ ni pa nujno res,
+ da so stekališča vsote dveh zaporedij paroma vsote stekališč teh dveh zaporedij.
+ Primer:
+
+\begin_inset Formula $a_{n}=\left(-1\right)^{n}$
+\end_inset
+
+ in
+\begin_inset Formula $b_{n}=-\left(-1\right)^{n}$
+\end_inset
+
+.
+ Njuni stekališči sta
+\begin_inset Formula $\left\{ -1,1\right\} $
+\end_inset
+
+,
+ toda
+\begin_inset Formula $a_{n}+b_{n}=0$
+\end_inset
+
+ ima le stekališče
+\begin_inset Formula $\left\{ 0\right\} $
+\end_inset
+
+,
+ ne pa tudi
+\begin_inset Formula $\left\{ 1,-1,2,-2\right\} $
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Corollary*
+sssssssssss
+\end_layout
+
+\begin_layout Corollary*
+sssssssssss
+\end_layout
+
+\begin_layout Corollary*
+sssssssssss
+\end_layout
+
+\begin_layout Corollary*
+sssssssssss
+\end_layout
+
+\begin_layout Corollary*
+sssssssssss
+\end_layout
+
+\begin_layout Corollary*
+sssssssssss
+\end_layout
+
+\begin_layout Corollary*
+sssssssssss
+\end_layout
+
+\begin_layout Corollary*
+sssssssssss
+\end_layout
+
+\begin_layout Corollary*
+sssssssssss
+\end_layout
+
+\end_body
+\end_document
diff --git a/šola/citati.bib b/šola/citati.bib
new file mode 100644
index 0000000..3c34475
--- /dev/null
+++ b/šola/citati.bib
@@ -0,0 +1,227 @@
+Osebna bibtex knjižnica citatov.
+Canonical: ~/projects/r/šola/citati.bib od 2024-07-28
+Stara različica je v ~/projects/sola-gimb-4/citati.bib
+
+@online{hpstrings,
+ author = {Nave, Carl Rod},
+ title = {{HyperPhysics Concepts: Standing Waves on a String}},
+ year = {2016},
+ url = {http://hyperphysics.phy-astr.gsu.edu./hbase/Waves/string.html},
+ urldate = {2016-11-09}
+}
+@online{cohen01,
+ author = {Cohen, Bram},
+ title = {BitTorrent - a new P2P app},
+ year = {2001},
+ url = {http://finance.groups.yahoo.com/group/decentralization/message/3160},
+ urldate = {2007-04-15},
+ note = "Internet Archive"
+}
+@online{harrison07,
+ author = {Harrison, David},
+ url = {http://www.bittorrent.org/beps/bep_0000.html},
+ urldate = {2023-02-28},
+ year = {2008},
+ title = {Index of BitTorrent Enhancement Proposals}
+}
+@online{norberg08,
+ author = {Loewenstern, Andrew and Norberg, Arvid},
+ url = {https://www.bittorrent.org/beps/bep_0005.html},
+ urldate = {2023-02-28},
+ year = {2020},
+ title = {DHT Protocol}
+}
+@online{jones15,
+ author = {Jones, Ben},
+ url = {https://torrentfreak.com/bittorrents-dht-turns-10-years-old-150607/},
+ urldate = {2023-02-28},
+ year = {2015},
+ title = {BitTorrent’s DHT Turns 10 Years Old}
+}
+@online{muo11,
+ author = {Mukherjee, Abhijeet},
+ url = {https://www.makeuseof.com/tag/btdigg-trackerless-torrent/},
+ urldate = {2023-02-28},
+ year = {2011},
+ title = {BTDigg: A Trackerless Torrent Search Engine}
+}
+@online{evseenko11,
+ author = {Evseenko, Nina},
+ url = {http://btdig.com},
+ urldate = {2023-02-28},
+ year = {2011},
+ title = {Btdigg BitTorrent DHT search engine}
+
+}
+@online{griffin17,
+ author = {Griffin, Andrew},
+ url = {https://www.independent.co.uk/tech/torrent-website-download-safe-legal-privacy-i-know-what-you-friends-spying-a7504266.html},
+ urldate = {2023-02-28},
+ year = {2017},
+ title = {'I Know What You Download': Website claims to let people see everything their friends have torrented}
+}
+@online{ikwyd,
+ url = {http://iknowwhatyoudownload.com},
+ urldate = {2023-02-28},
+ title = {I know what you download}
+}
+@online{cohen08,
+ author = {Choen, Bram},
+ url = {https://www.bittorrent.org/beps/bep_0003.html},
+ urldate = {2023-02-28},
+ year = {2017},
+ title = {The BitTorrent Protocol Specification}
+}
+@online{norberg09,
+ author = {Norberg, Arvid},
+ url = {https://www.bittorrent.org/beps/bep_0029.html},
+ urldate = {2023-02-28},
+ year = {2017},
+ title = {uTorrent transport protocol}
+}
+@online{hazel08,
+ author = {Hazel, Greg and Norberg, Arvid},
+ url = {https://www.bittorrent.org/beps/bep_0009.html},
+ urldate = {2023-02-28},
+ year = {2017},
+ title = {Extension for Peers to Send Metadata Files}
+}
+@online{strigeus08,
+ author = {Norberg, Arvid and Strigeus, Ludvig and Hazel, Greg},
+ url = {https://www.bittorrent.org/beps/bep_0010.html},
+ urldate = {2023-02-28},
+ year = {2017},
+ title = {Extension Protocol}
+}
+@online{v2,
+ author = {Cohen, Bram},
+ url = {https://www.bittorrent.org/beps/bep_0052.html},
+ urldate = {2023-02-28},
+ year = {2017},
+ title = {The BitTorrent Protocol Specification v2}
+}
+@incollection{maymounkov2002kademlia,
+ title={Kademlia: A peer-to-peer information system based on the xor metric},
+ author={Maymounkov, Petar and Mazieres, David},
+ booktitle={Peer-to-Peer Systems: First InternationalWorkshop, IPTPS 2002 Cambridge, MA, USA, March 7--8, 2002 Revised Papers},
+ pages={53--65},
+ year={2002},
+ publisher={Springer}
+}
+@inproceedings{pezoa2016foundations,
+ title={Foundations of JSON schema},
+ author={Pezoa, Felipe and Reutter, Juan L and Suarez, Fernando and Ugarte, Mart{\'\i}n and Vrgo{\v{c}}, Domagoj},
+ booktitle={Proceedings of the 25th International Conference on World Wide Web},
+ pages={263--273},
+ year={2016},
+ organization={International World Wide Web Conferences Steering Committee}
+}
+@online{harrison08,
+ author = {Harrison, David},
+ title = {Private Torrents},
+ year = {2008},
+ url = {https://www.bittorrent.org/beps/bep_0027.html},
+ urldate = {2023-02-28},
+}
+@techreport{Eastlake2001,
+ added-at = {2011-05-12T13:55:38.000+0200},
+ author = {Eastlake, D. and Jones, P.},
+ biburl = {https://www.bibsonomy.org/bibtex/2d2dcad25e57d68db18495f062e955cae/voj},
+ interhash = {276256be69453291def2a6fdbed1389d},
+ intrahash = {d2dcad25e57d68db18495f062e955cae},
+ keywords = {hash identifier sha1},
+ month = {9},
+ number = 3174,
+ organization = {IETF},
+ publisher = {IETF},
+ series = {RFC},
+ timestamp = {2013-01-06T13:13:36.000+0100},
+ title = {{US Secure Hash Algorithm 1 (SHA1)}},
+ type = {RFC},
+ year = 2001
+}
+@online{dis,
+ author = {Gams, Matjaž and others},
+ year = {2008},
+ title = {DIS Slovarček},
+ url = {https://dis-slovarcek.ijs.si/},
+ urldate = {2023-02-28}
+}
+@online{rhilip20,
+ author = {Rhilip},
+ year = {2020},
+ urldate = {2023-02-28},
+ url = {https://github.com/Rhilip/Bencode},
+ title = {A pure and simple PHP library for encoding and decoding Bencode data}
+}
+@conference{Kluyver2016jupyter,
+ Title = {Jupyter Notebooks -- a publishing format for reproducible computational workflows},
+ Author = {Thomas Kluyver and Benjamin Ragan-Kelley and Fernando P{\'e}rez and Brian Granger and Matthias Bussonnier and Jonathan Frederic and Kyle Kelley and Jessica Hamrick and Jason Grout and Sylvain Corlay and Paul Ivanov and Dami{\'a}n Avila and Safia Abdalla and Carol Willing},
+ Booktitle = {Positioning and Power in Academic Publishing: Players, Agents and Agendas},
+ Editor = {F. Loizides and B. Schmidt},
+ Organization = {IOS Press},
+ Pages = {87 - 90},
+ Year = {2016}
+}
+@online{sampleih,
+ author = {The 8472},
+ url = {https://www.bittorrent.org/beps/bep_0051.html},
+ title = {DHT Infohash Indexing},
+ urldate = {2023-02-28},
+ year = {2017}
+}
+@Article{Hunter:2007,
+ Author = {Hunter, J. D.},
+ Title = {Matplotlib: A 2D graphics environment},
+ Journal = {Computing in Science \& Engineering},
+ Volume = {9},
+ Number = {3},
+ Pages = {90--95},
+ abstract = {Matplotlib is a 2D graphics package used for Python for
+ application development, interactive scripting, and publication-quality
+ image generation across user interfaces and operating systems.},
+ publisher = {IEEE COMPUTER SOC},
+ doi = {10.1109/MCSE.2007.55},
+ year = 2007
+}
+@online{norberg14,
+ author = {Norberg, Arvid},
+ title = {DHT Security extension},
+ url = {https://www.bittorrent.org/beps/bep_0042.html},
+ urldate = {2023-02-28},
+ year = {2016}
+}
+@InProceedings{10.1007/3-540-45748-8_24,
+ author="Douceur, John R.",
+ editor="Druschel, Peter
+ and Kaashoek, Frans
+ and Rowstron, Antony",
+ title="The Sybil Attack",
+ booktitle="Peer-to-Peer Systems",
+ year="2002",
+ publisher="Springer Berlin Heidelberg",
+ address="Berlin, Heidelberg",
+ pages="251--260",
+ abstract="Large-scale peer-to-peer systems face security threats from faulty or hostile remote computing elements. To resist these threats, many such systems employ redundancy. However, if a single faulty entity can present multiple identities, it can control a substantial fraction of the system, thereby undermining this redundancy. One approach to preventing these ``Sybil attacks'' is to have a trusted agency certify identities. This paper shows that, without a logically centralized authority, Sybil attacks are always possible except under extreme and unrealistic assumptions of resource parity and coordination among entities.",
+ isbn="978-3-540-45748-0"
+}
+@book{cedilnik12,
+ place={Radovljica},
+ title={Matematični Priročnik},
+ publisher={Didakta},
+ author={Cedilnik, Anton and Razpet, Nada and Razpet, Marko},
+ year={2012}
+}
+@online{wikihashuniformity,
+ author="{Wikipedia contributors}",
+ title={Hash function},
+ url={https://w.wiki/An3L},
+ urldate={2024-07-29},
+ year={2024}
+}
+@online{maxmindgeoip2,
+ author={MaxMind},
+ title={GeoLite2 Free Geolocation Data},
+ url={https://dev.maxmind.com/geoip/geolite2-free-geolocation-data},
+ urldate={2024-07-31},
+ year={2024}
diff --git a/šola/članki/dht/.gitignore b/šola/članki/dht/.gitignore
new file mode 100644
index 0000000..abd26b3
--- /dev/null
+++ b/šola/članki/dht/.gitignore
@@ -0,0 +1,3 @@
+*.svg
+*.png
+*.tsv
diff --git a/šola/članki/dht/dokument.lyx b/šola/članki/dht/dokument.lyx
new file mode 100644
index 0000000..af668ed
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@@ -0,0 +1,2306 @@
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+\pdf_author "Your Name"
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+\shortcut idx
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+\end_header
+
+\begin_body
+
+\begin_layout Title
+Kaj prenašamo s protokolom BitTorrent?
+\end_layout
+
+\begin_layout Author
+Anton Luka Šijanec,
+
+\begin_inset CommandInset href
+LatexCommand href
+name "anton@sijanec.eu"
+target "anton@sijanec.eu"
+type "mailto:"
+literal "true"
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Institution
+Fakulteta za računalništvo in informatiko Univerze v Ljubljani
+\end_layout
+
+\begin_layout Abstract
+V članku predstavimo metodo za učinkovito in za omrežje neinvazivno metodo prenašanja metapodatkov iz pomožnega omrežja Kademlia mainline DHT protokola BitTorrent za izmenjavo datotek.
+ Sledi pregled/analiza z opisano metodo pridobljenih metapodatkov o datotekah na voljo v omrežju BitTorrent.
+\end_layout
+
+\begin_layout Abstract
+Porazdeljene razpršilne tabele (angl.
+ distributed hash table) so razpršilne tabele,
+ ki podatke,
+ ponavadi so to dokumenti,
+ strukturirani kot vrednost in njej pripadajoči ključ,
+ hranijo distribuirano na več vozliščih,
+ na katerih se podatki shranjujejo.
+ V računalniških sistemih se DHT uporablja za hrambo podatkov v omrežjih P2P (angl.
+ peer to peer),
+ kjer se podatki vseh uporabnikov enakomerno porazdelijo med vozlišča in so tako decentralizirani in preprosto dostopni članom omrežja.
+ Ker se podatki izmenjujejo znotraj omrežja na vozliščih,
+ ki z izvorom in destinacijo podatkov niso povezani,
+ jih lahko vozlišča v velikih količinah shranjujejo za potrebe statistične analize omrežja.
+\end_layout
+
+\begin_layout Abstract
+V raziskavi preverimo praktično zmožnost pridobivanja velike količine podatkov v omrežju BitTorrent za P2P izmenjavo datotek,
+ nato še analiziramo pridobljene podatke.
+ Vsaka poizvedba po seznamu imetnikov datotek vsebuje ključ podatka v DHT in se prenese preko okoli
+\begin_inset Formula $\log_{2}n$
+\end_inset
+
+ vozlišč,
+ kjer je
+\begin_inset Formula $n$
+\end_inset
+
+ število vseh uporabnikov v omrežju.
+ Ker vsaka poizvedba obišče tako veliko število vozlišč,
+ lahko med poizvedbo eno drugače nepovezano vozlišče prejme veliko obstoječih ključev v omrežju,
+ ki jih lahko uporabi za prenos metapodatkov v omrežju BitTorrent.
+\end_layout
+
+\begin_layout Abstract
+Osredotočili smo se le na na pridobivanje metapodatkov v omrežju BitTorrent,
+ samih datotek,
+ na katere se le-ti metapodatki sklicujejo in so v omrežju na voljo,
+ ker jih ponujajo drugi računalniki,
+ pa tako vsled tehničnih (njihove ogromne skupne velikosti) kot tudi pravnih razlogov (avtorsko zaščitena in protizakonita vsebina) nismo prenašali.
+ Metapodatki konceptualno sicer niso shranjeni v DHT (namesto metapodatkov o datotekah so v omrežju shranjeni seznami računalnikov,
+ od katerih si metapodatke lahko prenesemo),
+ vendar odkrivanje njihovega obstoja omogoči DHT.
+\end_layout
+
+\begin_layout Abstract
+S pridobljenimi metapodatki ugotovimo,
+ kateri odjemalci so najpopularnejši ter kakšna je razporeditev vsebine glede na tip datotek,
+ ki je na voljo preko protokola BitTorrent.
+\end_layout
+
+\begin_layout Keywords
+porazdeljena razpršilna tabela,
+ porazdeljeni sistemi,
+ omrežje P2P,
+\end_layout
+
+\begin_layout Keywords
+podatkovno rudarjenje,
+ BitTorrent
+\end_layout
+
+\begin_layout Section
+Introduction
+\end_layout
+
+\begin_layout Subsection
+Distribucija datotek po principu P2P
+\end_layout
+
+\begin_layout Standard
+Koncept P2P (angl.
+
+\shape italic
+peer-to-peer
+\shape default
+) predstavlja alternativen način distribucije identičnih datotek večim odjemalcem.
+ Namesto enega strežnika,
+ ki iste podatke odjemalcem pošlje vsakič znova,
+ v omrežjih P2P za distribucijo datotek vsak odjemalec podatke tako prejema kot tudi pošilja.
+ Odjemalec prejeto vsebino tudi sam deli drugim te vsebine željanim odjemalcem,
+ s čimer razbremeni ostale odjemalce.
+\end_layout
+
+\begin_layout Standard
+Odjemalec za druge izve s pomočjo centralnega strežnika ali pa drugačnega signalizacijskega protokola.
+ Ker se povezujejo neposredno,
+ medsebojno poznajo svoje internetne naslove.
+\end_layout
+
+\begin_layout Subsection
+Protokol BitTorrent
+\end_layout
+
+\begin_layout Standard
+Od 2008
+\begin_inset CommandInset citation
+LatexCommand cite
+key "harrison07"
+literal "false"
+
+\end_inset
+
+ je eden izmed popularnejših protokolov za P2P distribucijo BitTorrent,
+ razvit že 2001
+\begin_inset CommandInset citation
+LatexCommand cite
+key "cohen01"
+literal "false"
+
+\end_inset
+
+.
+ Zaradi razširljive zasnove ga je moč dopolnjevati —
+ dodajati nove funkcije.
+ Sprva je BitTorrent temeljil na centralnih strežnikih za koordinacijo rojev,
+ od leta 2005 pa z uvedbo protokola DHT lahko deluje povsem neodvisno.
+\begin_inset CommandInset citation
+LatexCommand cite
+key "jones15"
+literal "false"
+
+\end_inset
+
+
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+
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+
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+
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+
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+Razlaga
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+
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+\begin_inset Text
+
+\begin_layout Plain Layout
+odjemni program na računalniku,
+ za povezavo nanj potrebujemo IP naslov in vrata
+\end_layout
+
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+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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+
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+roj
+\begin_inset CommandInset citation
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+key "dis"
+literal "false"
+
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+
+
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+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
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+swarm
+\end_layout
+
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+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+več soležnikov,
+ ki prenašajo datoteke nekega torrenta
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+torrent/
+\begin_inset Newline newline
+\end_inset
+
+metainfo
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
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+\begin_inset Newline newline
+\end_inset
+
+metainfo
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+datoteka z metapodatki datotek;
+ imena,
+ velikosti,
+ zgoščene vrednosti in drugo
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+sledilnik
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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+
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+\end_layout
+
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+\begin_inset Text
+
+\begin_layout Plain Layout
+koordinacijski strežnik z naslovi soležnikov v rojih
+\end_layout
+
+\end_inset
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+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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+\end_layout
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+
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+delček datoteke konstantne dolžine
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+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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+
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+\end_layout
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+
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+
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+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+zgoščena vrednost serializiranih podatkov pod ključem
+\family typewriter
+info
+\family default
+ v torrentu,
+ ki unikatno opišejo ključne metapodatke o torrentu
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
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+objavi
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+
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+
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+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+pošiljanje obvestila v DHT ali na sledilnik,
+ da se odjemalec želi priključiti nekemu roju
+\end_layout
+
+\end_inset
+</cell>
+</row>
+</lyxtabular>
+
+\end_inset
+
+
+\begin_inset Caption Standard
+
+\begin_layout Plain Layout
+Slovar pojmov BitTorrenta
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Za prenos je treba poznati metapodatke o obstoječih datotekah,
+ ki so shranjeni v t.
+ i.
+ obliki .torrent,
+ strojno berljivi z bencoding serializirani datoteki.
+ Vsebujejo vsaj imena in poti datotek ter njihove zgoščene vrednosti,
+ ime torrenta,
+ lastnosti prenosa in velikost koščka.
+\end_layout
+
+\begin_layout Standard
+V raziskavi ne iščemo soležnikov s sledilniki in ne prenašamo datotek,
+ temveč samo prenašamo in analiziramo metapodatke.
+\end_layout
+
+\begin_layout Subsection
+Protokol Kademlia mainline DHT
+\end_layout
+
+\begin_layout Standard
+V BitTorrent je za iskanje soležnikov v roju uporabljen DHT (angl.
+
+\shape italic
+distributed hash table
+\shape default
+),
+ ki odpravi odvisnost od sledilnika.
+\end_layout
+
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+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+funkcija (XOR),
+ ki izrazi razdaljo med vozliščema kot 160-bitno številko
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+koš
+\begin_inset CommandInset citation
+LatexCommand cite
+key "dis"
+literal "false"
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+bucket
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+element usmerjevalne tabele,
+ ki glede na merilo razdalje vsebuje bližnja vozlišča
+\end_layout
+
+\end_inset
+</cell>
+</row>
+</lyxtabular>
+
+\end_inset
+
+
+\begin_inset Caption Standard
+
+\begin_layout Plain Layout
+Slovar pojmov Kademlie.
+ Slovenski prevodi niso ustaljeni.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Na visokem nivoju gre za abstraktno razpršilno tabelo,
+ shranjeno porazdeljeno na velikem omrežju vozlišč.
+ Podpira naslednji operaciji
+\begin_inset CommandInset citation
+LatexCommand cite
+key "norberg08"
+literal "false"
+
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Paragraph
+
+\family typewriter
+get_peers
+\end_layout
+
+\begin_layout Standard
+Vrne seznam soležnikov (IP naslov in vrata) za torrent,
+ opisan z njegovim infohashom.
+\end_layout
+
+\begin_layout Paragraph
+
+\family typewriter
+announce
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+qbittorrent pravi sporoči,
+ v deluge in transmission nisem našel prevoda
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+V seznam soležnikov za torrent,
+ opisan z njegovim infohashom,
+ vstavi IP naslov in vrata pošiljatelja zahteve.
+\end_layout
+
+\begin_layout Standard
+V raziskavi s sodelovanjem v DHT prestrezamo obstoječe ključe v razpršilni tabeli,
+ z njimi pridobimo soležnike,
+ od katerih prenesemo metapodatke o torrentih za kasnejšo analizo.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Float figure
+placement document
+alignment document
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\begin_inset Graphics
+ filename /root/projects/sola-gimb-4/inf/rn/predst/dht.svg
+ width 100col%
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+\begin_inset Caption Standard
+
+\begin_layout Plain Layout
+Shematski prikaz DHT
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Opis standardov
+\end_layout
+
+\begin_layout Paragraph
+Serializacija bkodiranje
+\family typewriter
+(bencoding)
+\end_layout
+
+\begin_layout Standard
+V BEP-0003
+\begin_inset CommandInset citation
+LatexCommand cite
+key "cohen08"
+literal "false"
+
+\end_inset
+
+ je opisan bencoding.
+ Z njim je serializirana večina struktur BitTorrenta in Kademlie.
+ Bkodiranje je podobno bolj znanemu JSONu
+\begin_inset CommandInset citation
+LatexCommand cite
+key "pezoa2016foundations"
+literal "false"
+
+\end_inset
+
+ —
+ vsebuje štiri podatkovne tipe:
+ niz,
+ število,
+ seznam in slovar.
+\end_layout
+
+\begin_layout Paragraph
+Datoteka metainfo/.torrent
+\end_layout
+
+\begin_layout Standard
+Za distribucijo vsebine s protokolom BitTorrent ustvarimo .torrent datoteko,
+ bkodiran slovar z metapodatki,
+ nujnimi za prenos datotek.
+ Za raziskavo so pomembni metapodatki pod ključem
+\family typewriter
+info
+\family default
+:
+\begin_inset CommandInset citation
+LatexCommand cite
+key "cohen08"
+literal "false"
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Itemize
+
+\family typewriter
+private
+\family default
+:
+ prepoved objavljanja preko DHT,
+ le preko sledilnikov (ti torrenti niso zajeti v raziskavi)
+\begin_inset CommandInset citation
+LatexCommand cite
+key "harrison08"
+literal "false"
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Itemize
+
+\family typewriter
+name
+\family default
+:
+ ime torrenta oz.
+ datoteke za enodatotečne torrente
+\end_layout
+
+\begin_layout Itemize
+
+\family typewriter
+piece length
+\family default
+:
+ velikost koščka —
+ datoteke so spojene skupaj in razdeljene na enako velike koščke
+\end_layout
+
+\begin_layout Itemize
+
+\family typewriter
+pieces
+\family default
+:
+ niz dolžine
+\begin_inset Formula $20n$
+\end_inset
+
+ (
+\begin_inset Formula $n$
+\end_inset
+
+ je število koščkov) s SHA-1 vrednostmi koščkov
+\begin_inset CommandInset citation
+LatexCommand cite
+key "Eastlake2001"
+literal "false"
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Itemize
+
+\family typewriter
+length
+\family default
+:
+ dolžina datoteke za enodatotečne torrente
+\end_layout
+
+\begin_layout Itemize
+
+\family typewriter
+files
+\family default
+:
+ seznam datotek v večdatotečnem torrentu —
+ vsaka datoteka je predstavljena kot slovar z
+\family typewriter
+length
+\family default
+ in
+\family typewriter
+path
+\family default
+.
+\end_layout
+
+\begin_layout Standard
+Kdor pozna infohash,
+ lahko od soležnika prenese metainfo in posledično tudi pripadajoče datoteke.
+ Roj najde in se vanj vključi s poizvedbo v DHT,
+ saj je infohash ključ v tej razpršilni tabeli
+\begin_inset CommandInset citation
+LatexCommand cite
+key "hazel08"
+literal "false"
+
+\end_inset
+
+.
+ Infohash običajno oblikujemo v t.
+ i.
+ magnetno povezavo (magnet URI):
+
+\family typewriter
+ magnet:?dn=
+\series bold
+ime torrenta
+\series default
+&xt=urn:btih:
+\series bold
+infohash
+\end_layout
+
+\begin_layout Standard
+Druga različica BitTorrenta ima drugačno metainfo strukturo s podobnimi podatki.
+ Uporablja SHA-256 in namesto
+\family typewriter
+pieces
+\family default
+ uporablja
+\family typewriter
+merkle hash tree
+\family default
+
+\begin_inset CommandInset citation
+LatexCommand cite
+key "v2"
+literal "false"
+
+\end_inset
+
+ za zgoščene vrednosti datotek.
+\end_layout
+
+\begin_layout Paragraph
+Graf DHT
+\end_layout
+
+\begin_layout Standard
+DHT vzdržuje sezname soležnikov v roju vseh obstoječih torrentov.
+ Vozlišča komunicirajo preko UDP in so del velikega usmerjenega grafa,
+ vsako s
+\begin_inset Formula $K\log_{2}n$
+\end_inset
+
+ (konstanta
+\begin_inset Formula $K=8$
+\end_inset
+
+,
+
+\begin_inset Formula $n$
+\end_inset
+
+ je število vseh vozlišč na svetu) povezavami —
+ vpisi v usmerjevalno tabelo.
+\end_layout
+
+\begin_layout Standard
+Vozlišče skrbi za urejeno usmerjevalno tabelo dosegljivih
+\begin_inset Foot
+status open
+
+\begin_layout Plain Layout
+dvosmerna komunikacija zaradi NAT ni samoumevna
+\end_layout
+
+\end_inset
+
+ vozlišč v koših;
+
+\begin_inset Formula $i$
+\end_inset
+
+ti koš hrani do
+\begin_inset Formula $K$
+\end_inset
+
+ med
+\begin_inset Formula $2^{i}$
+\end_inset
+
+ in
+\begin_inset Formula $2^{i-1}$
+\end_inset
+
+ po merilu XOR oddaljenih vozlišč,
+ torej je shranjenih veliko bližnjih in malo zelo oddaljenih vozlišč.
+\begin_inset CommandInset citation
+LatexCommand cite
+key "maymounkov2002kademlia"
+literal "false"
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Paragraph
+Poizvedbe po grafu
+\end_layout
+
+\begin_layout Standard
+Sprehod po grafu med poljubnima vozliščema je torej dolg v povprečju
+\begin_inset Formula $\log n$
+\end_inset
+
+ (
+\begin_inset Formula $n$
+\end_inset
+
+ kot prej).
+ Roj torrenta z infohashom
+\begin_inset Formula $x$
+\end_inset
+
+ je shranjen na vozliščih z ID blizu
+\begin_inset Formula $x$
+\end_inset
+
+,
+ tedaj ima poizvedba po soležnikih/objavljanje soležnika časovno kompleksnost
+\begin_inset Formula $O\left(\log n\right)$
+\end_inset
+
+.
+ Za pridobitev seznama soležnikov torrenta pošljemo bkodiran UDP paket tipa
+\family typewriter
+get_peers
+\family default
+
+\begin_inset Formula $t$
+\end_inset
+
+
+\begin_inset Foot
+status open
+
+\begin_layout Plain Layout
+odvisno od implementacije
+\end_layout
+
+\end_inset
+
+ vozliščem iz usmerjevalne tabele,
+ ki so blizu infohasha.
+ Pozvana vozlišča odgovorijo s seznamom do
+\begin_inset Formula $K$
+\end_inset
+
+ temu infohashu najbližjih vozlišč in seznamom soležnikov za ta torrent,
+ če ga imajo.
+ Novodobljenim vozliščem spet pošljemo poizvedbo
+\family typewriter
+get_peers
+\family default
+ in postopek nadaljujemo,
+ dokler ne najdemo nekaj infohashu najbližjih vozlišč.
+ V tista pošiljamo objave in od njih še naprej prejemamo informacije o roju.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Float figure
+placement document
+alignment document
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\align center
+\begin_inset Graphics
+ filename Dht_example_SVG.svg
+ width 50col%
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+\begin_inset Caption Standard
+
+\begin_layout Plain Layout
+\begin_inset CommandInset label
+LatexCommand label
+name "fig:Usmerjevalna-tabela-za"
+
+\end_inset
+
+Usmerjevalna tabela za vozlišče 110 (koši so osenčeni)
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Metode
+\end_layout
+
+\begin_layout Standard
+Vsaka poizvedba obišče
+\begin_inset Formula $\log n$
+\end_inset
+
+ vozlišč,
+ torej vsako vozlišče v DHT prejema ogromno ključev —
+ infohashov.
+ V raziskavi v C spišemo program travnik,
+ nepopolno implementacijo odjemalca BitTorrent s poudarkom na DHT.
+ Osredotočimo se na zajem metapodatkov iz ključev,
+ ki jih prejmemo s sodelovanjem v omrežju.
+\end_layout
+
+\begin_layout Standard
+Ko vozlišče,
+ na katerem teče travnik,
+ prejme paket z infohashom,
+ ga doda v seznam željenih torrentov.
+ Neprestano posodablja roje torrentov znanih infohashov in se povezuje na soležnike iz njih,
+ dokler mu ne uspe prenesti metapodatkov torrenta oziroma dokler ne obupa/preteče 256 sekundni TTL torrenta.
+ Izdeluje .torrent datoteke z najdenimi metapodatki in internetnim naslovom ter ime programske opreme soležnika,
+ od katerega je metapodatke prejel.
+ Ne objavlja se v roj,
+ ker ne redistribuira niti metapodatkov niti datotek.
+\end_layout
+
+\begin_layout Standard
+Da program prvič začne sodelovati z omrežjem —
+ da ga sosednja vozlišča vpišejo v svoje usmerjevalne tabele —
+ prenese metapodatke vgrajenega torrenta
+\family typewriter
+Big Buck Bunny
+\family default
+.
+\end_layout
+
+\begin_layout Standard
+Za implementacijo spišemo knjižnico za bkodiranje in bdekodiranje,
+ knjižnico za DHT in nekaj funkcij za prenos metapodatkov od soležnikov preko TCP.
+\end_layout
+
+\begin_layout Standard
+Za obdelavo dobljenih torrent datotek uporabimo Jupyter Notebook
+\begin_inset CommandInset citation
+LatexCommand cite
+key "Kluyver2016jupyter"
+literal "false"
+
+\end_inset
+
+ in spišemo preprosto knjižnico za razčlenjevanje .torrent metainfo datotek,
+ ki jih generira travnik.
+\end_layout
+
+\begin_layout Section
+Rezultati
+\end_layout
+
+\begin_layout Subsection
+Zajem
+\end_layout
+
+\begin_layout Standard
+Podatke smo zajemali iz različnih lokacij in v različnih časovnih obdobjih:
+\end_layout
+
+\begin_layout Itemize
+januarja in februarja 2023:
+\end_layout
+
+\begin_deeper
+\begin_layout Itemize
+16 dni:
+ domači optični priključek v Sloveniji (T-2):
+ 47863 torrentov (3084321 datotek,
+ 259 TiB,
+ 29 sekund na torrent)
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+travnik
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Itemize
+31 dni VPS v Grčiji (grNet)
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+1
+\end_layout
+
+\end_inset
+
+:
+
+\family roman
+\series medium
+\shape up
+\size normal
+\emph off
+\nospellcheck off
+\bar no
+\strikeout off
+\xout off
+\uuline off
+\uwave off
+\noun off
+\color none
+412846
+\family default
+\series default
+\shape default
+\size default
+\emph default
+\nospellcheck default
+\bar default
+\strikeout default
+\xout default
+\uuline default
+\uwave default
+\noun default
+\color inherit
+ torrentov (17101702 datotek,
+ 1881 TiB,
+ 6 sekund na torrent)
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+okeanos
+\end_layout
+
+\end_inset
+
+
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+XX dni VPS v Grčiji (grNet) 2:
+ 342220 torrentov ()
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+oliwerix
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_deeper
+\begin_layout Itemize
+5 dni junija 2024 na domačem optičnem priključku v Sloveniji (T-2):
+ 62110 torrentov (3725125 datotek,
+ 345 TiB,
+ 7 sekund na torrent)
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+2024b
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsubsection
+Primer strukture torrent datoteke z metapodatki
+\end_layout
+
+\begin_layout Standard
+Spodaj je iz bencoding v JSON pretvorjena metainfo datoteka prevzetega torrenta z infohashom
+\family typewriter
+696802a16728636cd72617e4cd7b64e3ca314e71
+\family default
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+lstinputlisting[language=json,firstnumber=1,
+ numbers=none,
+ breaklines=true,
+ basicstyle=
+\backslash
+tiny]{../../../../sola-gimb-4/inf/rn/predst/torrent.json}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsection
+Analiza
+\end_layout
+
+\begin_layout Subsubsection
+Odjemalci,
+ od katerih so bili prejeti torrenti
+\end_layout
+
+\begin_layout Standard
+Imenom programom odstranimo različico in jim ročno normaliziramo ime
+\begin_inset Foot
+status open
+
+\begin_layout Plain Layout
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+textmu Torrent
+\end_layout
+
+\end_inset
+
+ se sicer pojavi dvakrat,
+ enkrat ima znak mikro,
+ enkrat pa grško črko mu.
+ Unicode namreč ta dva znaka,
+ ki sicer izgledata identično,
+ hrani pod dvema različnima kodama.
+\end_layout
+
+\end_inset
+
+ ter prikažemo njihovo gostoto v populaciji.
+\begin_inset CommandInset citation
+LatexCommand cite
+key "Hunter:2007"
+literal "false"
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Float figure
+placement document
+alignment document
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\begin_inset Graphics
+ filename /root/projects/sola-gimb-4/inf/rn/dok/odjemalci_1_ods.png
+ width 100col%
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+\begin_inset Caption Standard
+
+\begin_layout Plain Layout
+Reprezentacija odjemalcev,
+ ki predstavljajo vsaj en odstotek populacije,
+ na logaritemski skali
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsubsection
+Različice odjemalca
+\family typewriter
+qBittorrent
+\family default
+ skozi čas
+\end_layout
+
+\begin_layout Standard
+Primerjava porazdelitve različic po zgornji analizi najpopularnejšega odjemalca na
+\begin_inset Formula $\log_{10}$
+\end_inset
+
+ skali pokaže višanje različic skozi čas.
+
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+Različice smo razvrstili s pythonsko funkcijo
+\family typewriter
+packaging.version.Version
+\family default
+.
+\end_layout
+
+\end_inset
+
+ V obeh letih smo prejeli torrente od skupno 88 različnih inačic qBittorrenta.
+ V 2023 smo največ torrentov prejeli od odjemalcev različice 4.5.0,
+ v 2024 pa od odjemalcev različice 4.6.3.
+
+\begin_inset Float figure
+placement document
+alignment document
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+\begin_inset Graphics
+ filename /root/projects/r/šola/članki/dht/verzije2324.png
+ width 100col%
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\begin_inset Graphics
+ filename /root/projects/r/šola/članki/dht/verzije2324promil.png
+ width 100col%
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+\begin_inset Caption Standard
+
+\begin_layout Plain Layout
+Primerjava distribucij različic odjemalca
+\family typewriter
+qBittorrent
+\family default
+ med 2023 (plavo) in 2024 (roza),
+ ki predstavljajo vsaj promil populacije (delež).
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsubsection
+Geolokacija IP naslovov odjemalcev
+\end_layout
+
+\begin_layout Standard
+Z uporabo podatkovne zbirke MaxMind GeoLite2
+\begin_inset CommandInset citation
+LatexCommand cite
+key "maxmindgeoip2"
+literal "false"
+
+\end_inset
+
+ IP naslovom,
+ od katerih smo prejeli torrente,
+ določimo izvirno državo.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Float figure
+placement document
+alignment document
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\begin_inset Graphics
+ filename countries_procent.png
+ width 100col%
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+\begin_inset Caption Standard
+
+\begin_layout Plain Layout
+Reprezentativnost držav,
+ iz katerih smo prenesli metainfo,
+ na linearni skali.
+ Prikazane so le države,
+ iz katerih izvira vsaj odstotek populacije.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsubsection
+Predstavnost ključev v prejetih slovarjih
+\family typewriter
+info
+\end_layout
+
+\begin_layout Standard
+Poleg standardnih obveznih nekateri torrenti vsebujejo tudi dodatne metapodatke v slovarju info.
+ Pogostost slednjih prikazuje spodnji grafikon.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Float figure
+placement document
+alignment document
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\begin_inset Graphics
+ filename /root/projects/sola-gimb-4/inf/rn/dok/vsi_ključi.png
+ width 100col%
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+\begin_inset Caption Standard
+
+\begin_layout Plain Layout
+Reprezentacija ključev v slovarju
+\family typewriter
+info
+\family default
+ na logaritemski skali
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsubsection
+Tipi datotek,
+ ki se prenašajo v torrentih
+\end_layout
+
+\begin_layout Standard
+Iz končnice datoteke izvemo tip datoteke.
+ Vsakemu torrentu priredimo reprezentativen tip,
+ tisti,
+ ki po velikosti prevladuje v torrentu.
+ Glede na število torrentov z nekim reprezentativnim tipom kvantificiramo pogostost tega datotečnega tipa za tipe,
+ ki zavzemajo vsaj promil populacije.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Float figure
+placement document
+alignment document
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\begin_inset Graphics
+ filename /root/projects/sola-gimb-4/inf/rn/dok/reprezentativni_.1_ods.png
+ width 100col%
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+\begin_inset Caption Standard
+
+\begin_layout Plain Layout
+Reprezentativni tipi torrentov,
+ ki predstavljajo vsaj en promil populacije,
+ na logaritemski skali
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Razvidno je,
+ da je večina torrentov namenjena prenosu videovsebin,
+ zvočnih datotek in stisnjenih arhivov.
+\end_layout
+
+\begin_layout Standard
+Če bi za določilo pojavnosti tipa uporabili število datotek,
+ bi prevladovali tipi vsebin,
+ ki so ponavadi preneseni kot kopica datotek,
+ denimo slike (diagram v prilogi na sliki
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "fig:Pojavnost-tipa-kot"
+plural "false"
+caps "false"
+noprefix "false"
+nolink "false"
+
+\end_inset
+
+),
+ če pa bi za določilo pojavnosti tipa uporabili velikost datotek tega tipa,
+ pa bi prevladovali tisti tipi,
+ ki zasedajo več prostora.
+ V tem primeru bi npr.
+ videovsebine zaradi svoje velikosti občutno presegale digitalne knjige (diagram v prilogi na sliki
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "fig:Pojavnost-tipa-kot-velikost"
+plural "false"
+caps "false"
+noprefix "false"
+nolink "false"
+
+\end_inset
+
+).
+\end_layout
+
+\begin_layout Standard
+\begin_inset Note Note
+status open
+
+\begin_layout Subsubsection
+\begin_inset CommandInset label
+LatexCommand label
+name "subsec:Porazdeljenost-infohashov"
+
+\end_inset
+
+Porazdeljenost infohashov
+\end_layout
+
+\begin_layout Plain Layout
+Zaradi delovanja poizvedb v DHT pričakujemo,
+ da je porazdelitev infohashov po celotnem sprektru števil z intervala
+\begin_inset Formula $\left[0,2^{160}-1\right]$
+\end_inset
+
+ gostejša okoli IDja vozlišča,
+ ki ga je med prenašanjem imelo naše iskalno vozlišče.
+ IDje smo izbrali naključno na vsaki merilni lokaciji in jih med meritvijo tudi nekajkrat zamenjali,
+ zato spodnjem grafikonu opazimo vrhove tam,
+ kjer so bili naši IDji med zajemom podatkov.
+\end_layout
+
+\begin_layout Plain Layout
+\begin_inset Float figure
+placement document
+alignment document
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\begin_inset Caption Standard
+
+\begin_layout Plain Layout
+Razporeditev pridobljenih infohashov na spektru vseh infohashov
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsection
+Diskusija
+\end_layout
+
+\begin_layout Paragraph
+Statistična kvaliteta vzorca
+\end_layout
+
+\begin_layout Standard
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+V razdelku
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "subsec:Porazdeljenost-infohashov"
+plural "false"
+caps "false"
+noprefix "false"
+nolink "false"
+
+\end_inset
+
+ opazimo neenakomerno porazdeljenost infohashov,
+ kar je posledica načina vzorčenja torrentov.
+ Vsa populacija ima namreč zaradi delovanja zgoščevalne funkcije SHA-1 homogeno porazdeljene infohashe.
+ Kljub temu trdimo,
+ da pridobljen vzorec torrentov dobro predstavlja celotno populacijo.
+
+\end_layout
+
+\end_inset
+
+Zaradi lastnosti uniformne porazdelitve zgoščevalne funkcije
+\begin_inset CommandInset citation
+LatexCommand cite
+key "wikihashuniformity"
+literal "false"
+
+\end_inset
+
+ mesto infohasha na intervalu vseh možnih infohashov ni odvisno od metapodatkov.
+ Kot posledico načina vzorčenja z DHT pričakujemo,
+ da je porazdelitev infohashov prejetih torrentov po celotnem sprektru števil z intervala
+\begin_inset Formula $\left[0,2^{160}-1\right]$
+\end_inset
+
+ gostejša okoli IDja vozlišča,
+ ki ga je med prenašanjem imelo naše iskalno vozlišče.
+ IDje smo tekom raziskave izbrali naključno na vsaki merilni lokaciji in jih med meritvijo tudi nekajkrat zamenjali.
+ Kljub temu je vsled nepovezanosti vsebine in infohasha vzorec še vedno statistično reprezentativen.
+ Zajem ne more biti pristranski glede na metapodatke,
+ ker nikjer v procesu zajema ne obravnavamo torrentov glede na metainfo,
+ temveč le glede na infohash.
+\end_layout
+
+\begin_layout Paragraph
+Težava z zajemom podatkov
+\end_layout
+
+\begin_layout Standard
+Vsled majhne velikosti UDP paketov DHT glavno ozko grlo pri zajemu predstavlja število paketov,
+ ki jih mrežna oprema lahko posreduje v sekundi.
+ Domača optična povezava dopušča do okoli 2000 paketov na sekundo na naključno porazdeljene IP naslove,
+ odjemno mesto na VPS pa je imelo to omejitev veliko višjo,
+ zato smo tam v istem časovnem intervalu shranili veliko več torrent datotek.
+\end_layout
+
+\begin_layout Paragraph
+Etičnost in legitimnost rudarjenja podatkov
+\end_layout
+
+\begin_layout Standard
+Čeprav gre za izrazito osebne podatke,
+ se morajo uporabniki BitTorrent omrežja zavedati,
+ da so njihovi prenosi
+\shape italic
+a priori
+\shape default
+ javni,
+ tudi če jih nihče aktivno ne zajema.
+ Nekateri BitTorrent odjemalci uporabnike ob prvem zagonu o tem obvestijo.
+\end_layout
+
+\begin_layout Section
+Priloge
+\end_layout
+
+\begin_layout Standard
+Izvorna koda programa travnik in ipynb datotek za analizo podatkov je na voljo na
+\begin_inset CommandInset href
+LatexCommand href
+name "http://ni.šijanec.eu./sijanec/travnik"
+target "http://ni.šijanec.eu./sijanec/travnik"
+literal "false"
+
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Korpus zajetih metapodatkov je na voljo na
+\begin_inset CommandInset href
+LatexCommand href
+name "rsync://b.sjanec.eu./travnik"
+target "rsync://b.sijanec.eu./travnik"
+type "other"
+literal "false"
+
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Float figure
+placement document
+alignment document
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\begin_inset Graphics
+ filename po_številu_datotek.png
+ width 100col%
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+\begin_inset Caption Standard
+
+\begin_layout Plain Layout
+\begin_inset CommandInset label
+LatexCommand label
+name "fig:Pojavnost-tipa-kot"
+
+\end_inset
+
+Pojavnost tipa kot število datotek
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Float figure
+placement document
+alignment document
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\begin_inset Graphics
+ filename /root/projects/sola-gimb-4/inf/rn/dok/po_velikosti_datotek.png
+ width 100col%
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+\begin_inset Caption Standard
+
+\begin_layout Plain Layout
+\begin_inset CommandInset label
+LatexCommand label
+name "fig:Pojavnost-tipa-kot-velikost"
+
+\end_inset
+
+Pojavnost tipa kot velikost datotek (tipi,
+ ki zavzamejo vsaj odstotek populacije)
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset CommandInset bibtex
+LatexCommand bibtex
+btprint "btPrintCited"
+bibfiles "/root/projects/r/šola/citati"
+options "IEEEtran"
+encoding "default"
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section*
+Viri slik
+\end_layout
+
+\begin_layout Itemize
+Slika
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "fig:Usmerjevalna-tabela-za"
+plural "false"
+caps "false"
+noprefix "false"
+nolink "false"
+
+\end_inset
+
+:
+ Limaner:
+ nespremenjena,
+ izvorna pod CC BY-SA
+\end_layout
+
+\begin_layout Section*
+Dovoljenje
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+doclicenseImage[imagewidth=2cm]
+\end_layout
+
+\end_inset
+
+
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+TODO:
+ preštej datoteke v oliwerix,
+ še enkrat nariši vse grafe upoštevajoč vse torrente,
+ primerjaj verzije med travnik in 2024b
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_body
+\end_document
generated by cgit v1.2.3 (git 2.25.1) at 2024-11-09 00:29:52 +0100