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\QTR{small}{Matematick\U{e1} anal\U{fd}za I online - D\U{f4}kazy\dotfill \thepage }}
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\begin{document}
\author{A. U. Thor}
\title{Lab Report}
\date{The Date }
\maketitle
\begin{abstract}
A Laboratory report created with Scientific Notebook
\end{abstract}
\section{Postupnosti a rady re\'{a}lnych \v{c}\'{\i}sel}
\subsection{D\^{o}kaz vety}
\textbf{D\^{o}kaz: }Nech $\left\{ a_{n}\right\} _{n=1}^{\infty }$ je ohrani%
\v{c}en\'{a} a neklesaj\'{u}ca (pre nerast\'{u}cu, rast\'{u}cu a klesaj\'{u}%
cu postupnos\v{t} je d\^{o}kaz podobn\'{y}). Preto\v{z}e $\left\{
a_{n}\right\} _{n=1}^{\infty }$ je ohrani\v{c}en\'{a} $\left| a_{n}\right|
\leq M,\,\forall n\in \mathbf{N}$. Ozna\v{c}me $S=\left\{
a_{1},\,a_{2},\,...,\,a_{n},\,...\right\} $ a uva\v{z}ujme ju ako mno\v{z}%
inu. Nech $L=\sup S$ ($L$ existuje). Uk\'{a}\v{z}eme, \v{z}e $%
\lim_{n\longrightarrow \infty }a_{n}=L$. Nech $\varepsilon >0$. Preto\v{z}e $%
L=\sup S$, potom $L-\varepsilon $ nie je horn\'{y}m ohrani\v{c}en\'{\i}m mno%
\v{z}iny $S$. Existuje teda $a_{n_{0}}\in S$, tak\'{e} \v{z}e
\[
L-\varepsilon