\documentclass{article} \usepackage{amssymb} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %TCIDATA{OutputFilter=LATEX.DLL} %TCIDATA{Created=Monday, June 25, 2001 17:45:57} %TCIDATA{LastRevised=Friday, January 24, 2003 23:46:26} %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{Komplexn\'{e} \v{c}\'{\i}sla a funkcie komplexnej premennej} \subsection{D\^{o}kaz vety} \textbf{D\^{o}kaz: }a) Ak $\lim_{n\longrightarrow \infty }z_{n}=z_{0},$ potom pre ka\v{z}d\'{e} $\varepsilon >0$ existuje $N(\varepsilon )$ tak\'{e}% , \v{z}e $|z_{n}-z_{0}|<\varepsilon $ t.j. \[ |(x_{n}-x_{0})+i(y_{n}-y_{0})|=\sqrt{(x_{n}-x_{0})^{2}+(y_{n}-y_{0})^{2}}% <\varepsilon . \]% Preto\v{z}e \[ |x_{n}-x_{0}|\leq |z_{n}-z_{0}|<\varepsilon \text{ \ a \ }|y_{n}-y_{0}|\leq |z_{n}-z_{0}|<\varepsilon \]% pre $n>N(\varepsilon ),$ potom $\lim_{n\longrightarrow \infty }x_{n}=x_{0}$ a $\lim_{n\longrightarrow \infty }y_{n}=y_{0}$. b) Nech $\lim_{n\longrightarrow \infty }x_{n}=x_{0}$ a $\lim_{n% \longrightarrow \infty }y_{n}=y_{0}$ a nech $\varepsilon >0$ je \v{l}ubovo% \v{l}n\'{e}. Potom existuj\'{u} $N_{1}$ a $N_{2}$ tak\'{e}, \v{z}e pre ka% \v{z}d\'{e} $n>N_{1}$ plat\'{\i} $|x_{n}-x_{0}|<\frac{\varepsilon }{\sqrt{2}} $ a pre ka\v{z}d\'{e} $n>N_{2}$ plat\'{\i} $|y_{n}-y_{0}|<\frac{\varepsilon }{\sqrt{2}}.$ Nech $N=\max (N_{1},N_{2}).$ Potom pre ka\v{z}d\'{e} $n>N$ m% \'{a}me \[ |z_{n}-z_{0}|=\sqrt{(x_{n}-x_{0})^{2}+(y_{n}-y_{0})^{2}}<\varepsilon , \]% teda \[ \lim_{n\longrightarrow \infty }z_{n}=z_{0}.\blacksquare \] \begin{center} \begin{tabular}{|c|} \hline {\small \hyperref{Sp\"{a}\v{t}}{}{}{K12.tex#3}} \\ \hline \end{tabular} \end{center} \rule{6.5in}{0.04in} \textsl{Matematick\'{a} anal\'{y}za III} \end{document}