\documentclass{article} \usepackage{amssymb} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %TCIDATA{OutputFilter=LATEX.DLL} %TCIDATA{Created=Monday, June 25, 2001 17:45:57} %TCIDATA{LastRevised=Friday, January 24, 2003 22:37:33} %TCIDATA{} %TCIDATA{} %TCIDATA{CSTFile=On line bluem.cst} %TCIDATA{PageSetup=72,72,72,72,0} %TCIDATA{AllPages= %H=36 %F=36,\PARA{038
\QTR{small}{Matematick\U{e1} anal\U{fd}za III online - D\U{f4}kazy\dotfill \thepage }} %} \newtheorem{theorem}{Theorem} \newtheorem{acknowledgement}[theorem]{Acknowledgement} \newtheorem{algorithm}[theorem]{Algorithm} \newtheorem{axiom}[theorem]{Axiom} \newtheorem{case}[theorem]{Case} \newtheorem{claim}[theorem]{Claim} \newtheorem{conclusion}[theorem]{Conclusion} \newtheorem{condition}[theorem]{Condition} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{criterion}[theorem]{Criterion} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{exercise}[theorem]{Exercise} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{notation}[theorem]{Notation} \newtheorem{problem}[theorem]{Problem} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \newtheorem{solution}[theorem]{Solution} \newtheorem{summary}[theorem]{Summary} \newenvironment{proof}[1][Proof]{\textbf{#1.} }{\ \rule{0.5em}{0.5em}} \input{tcilatex} \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{Analytickos\v{t} s\'{u}\v{c}tu mocninov\'{e}ho radu} \subsection{D\^{o}kaz vety} \textbf{D\^{o}kaz: }Z \hyperref{vety}{}{}{K13.tex#8} plynie, \v{z}e $f\left( z\right) $ je spojit\'{a} funkcia na $C,$ \v{c}o znamen\'{a}, \v{z}e integr% \'{a}l je definovan\'{y} (existuje). Ozna\v{c}me \begin{equation} r_{n}=\int_{C}f\left( z\right) dz-\sum_{k=1}^{n}\int_{C}f_{k}\left( z\right) dz=\int_{C}\left[ f\left( z\right) -\sum_{k=1}^{n}f_{k}\left( z\right) % \right] dz\, \tag{(i)} \end{equation}% Preto\v{z}e rad $\sum_{n=1}^{\infty }f_{n}\left( z\right) $ je rovnomerne konvergentn\'{y} na $C,$ potom pre ka\v{z}d\'{e} $\varepsilon >0$ existuje $% n_{0}\in \mathbf{N}$ tak\'{e}, \v{z}e pre ka\v{z}d\'{e} $n\geq n_{0}$ a ka% \v{z}d\'{e} \[ z\in C,\,\left| f\left( z\right) -\sum_{k=1}^{n}f_{k}\left( z\right) \right| <\varepsilon \]% potom \[ \left| r_{n}\right| =\left| \int_{C}\left[ f\left( z\right) -\sum_{k=1}^{n}f_{k}\left( z\right) \right] dz\right| \leq \int_{C}\left| % \left[ f\left( z\right) -\sum_{k=1}^{n}f_{k}\left( z\right) \right] \right| dz\leq \varepsilon d(C), \]% t.j. $\lim_{n\longrightarrow \infty }r_{n}=0$ a (i) plat\'{\i}. $% \blacksquare $ \begin{center} \begin{tabular}{|c|} \hline {\small \hyperref{Sp\"{a}\v{t}}{}{}{K33.tex#4}} \\ \hline \end{tabular} \end{center} \rule{6.5in}{0.04in} \textsl{Matematick\'{a} anal\'{y}za III} \end{document}