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%%  Example 04-03-1 on page 75.
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%%  Copyright (C) 2012 Vo\ss 
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%%  of the LaTeX Project Public License, either version 1.3
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\PassOptionsToClass{}{beamer}
\documentclass{exabeamer}
\usepackage[utf8]{inputenc}

%\StartShownPreambleCommands
\mode<article>{% only article mode
  \usepackage{fullpage} \usepackage[linktocpage]{hyperref} }
\mode<presentation>{% only slides
  \setbeamertemplate{background canvas}[vertical shading][bottom=red!10,top=blue!10]
  \usetheme{Warsaw} \usefonttheme[onlysmall]{structurebold} }
%\StopShownPreambleCommands

\begin{document}
\title{Introduction to analytic geometry}
\author{Gerhard Kowalewski}  \date{1910}
\frame{\titlepage}
\section<presentation>*{Overview}
\begin{frame}{Overview}  \tableofcontents[part=1,pausesections] \end{frame}

\AtBeginSubsection[]{\begin{frame}<beamer>
    \frametitle{Overview} \tableofcontents[current,currentsubsection] \end{frame} }

\part<presentation>{Main part}

\section{Research and studies}
\begin{frame}{The integral and its geometric applications.}
We assume that the theory of irrational numbers is known.
\end{frame}
\subsection{Interval}
\begin{frame}{Definition}
The \emph{interval} $\langle a,b\rangle$ consists of all numbers $x$ that
satisfy the condition $a\le x\le b$.
\end{frame}
\subsection{Sequence of numbers}
\begin{frame}{Definition of a sequence}
A \emph{sequence of numbers} or \emph{sequence} is created by replacing each member
of the infinite sequence of numbers $1,2,3,\ldots$ by some rational or irrational
number, i.e.\ each $n$ by a number $x_n$.
\end{frame}
\subsection{Limits}
\begin{frame}{Definition of a limit}
$\lim x_n=g$ means that almost all members of the sequence are within each neighbourhood of $g$.
\end{frame}
\subsection{Convergence criterion}
\begin{frame}{Definition of convergence}
\textbf{Convergence criterion}: The sequence $x_1,x_2,x_3,\ldots$ converges if and
only if \textbf{each} sub-sequence $x^\prime_1,x^\prime_2, x^\prime_3,\ldots$
satisfies the relation $\lim(x_n-x^\prime_n)=0$.
\end{frame}
\end{document}