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\begin{document}
\title{Introduction to Analytic Geometry}
\author{Gerhard Kowalewski}
\date{1910}
\frame{\maketitle}
\section{Research and studies}
\begin{frame}{The integral and its geometric applications.}
We assume that the theory of irrational numbers is known.

\begin{enumerate}
 \item The \emph{interval} $\langle a,b\rangle$ consists of all numbers $x$
    that satisfy the condition $a\le x\le b$.
 \item 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 $x_n$.
 \item $\lim x_n=g$ means that almost all members of the sequence are within each
    neighbourhood of $g$.
 \item \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{enumerate}
\end{frame}
\end{document}
