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\begin{document}
\begin{frame}{Negative example}
We assume that the \textbf{theory} of \textbf{irrational numbers} is known.
\begin{enumerate}
 \item The \textbf{interval} $\langle a,b\rangle$ contains all \textbf{numbers} $x$ that satisfy
   the condition $a\le x \le b$.
 \item A \textbf{sequence of numbers} or \textbf{sequence} is created by replacing each
       member of the infinite \textbf{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
   \textbf{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}
\begin{frame}{Positive example}
We assume that the \emph{theory} of \emph{irrational numbers} is know.
\begin{enumerate}
 \item The \emph{interval} $\langle a,b\rangle$ contains all \emph{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 \emph{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
   \emph{neighbourhood} of $g$.
 \item  \emph{Convergence criterion}: The sequence $x_1,x_2,x_3,\ldots$ converges
if and only if \emph{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}
