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\author{C. Desjardins and J. Lewis}
\title{Cauchy Sequences and Convergence}
\date{October 19, 2009}
\begin{document}
\maketitle
\section{Introduction and definitions}
In this paper, we present the proof of a theorem relating convergent sequences to Cauchy sequences. This theorem helps motivate the idea of a complete metric space and of the notion of topological completion. We begin by presenting the definitions we need to state the theorem.
\begin{dfn}
A sequence $(s_n)_{n \in \N}$ in a metric space $(X, d)$ is said to be a \emph{Cauchy sequence} if for all $\eps > 0$ there exists $N \in \N$ such that for all $m, n > N$ we have $d(s_n, s_m) < \eps$.
\end{dfn}
\begin{dfn}
A sequence $(s_n)_{n \in \N}$ in a metric space $(X, d)$ is said to \emph{converge to} $L \in X$ if for all $\eps > 0$ there exists $N \in \N$ such that for all $n > N$ we have $d(s_n, L) < \eps$. If there exists $L \in X$ such that $(s_n)_{n \in \N}$ converges to $L$ then we say that $(s_n)_{n \in \N}$ \emph{converges} or \emph{is convergent}.
\end{dfn}
With these definitions in hand, we proceed to our main result.
\section{Main Theorem}
\begin{thm}
Every convergent sequence $(s_n)_{n \in \N}$ in the metric space $(X, d)$ is a Cauchy sequence.
\end{thm}
\begin{proof}
Suppose that $(s_n)_{n \in \N}$ is a convergent sequence. Then it has a limit $L$. By choosing a small $\eps$, we have by the definition of convergence that eventually all terms of $(s_n)_{n \in \N}$ will be arbitrarily to $L$. Then choosing any two of these terms, say $s_n$ and $s_m$, and applying the triangle inequality, we have that
\[
d(s_m, s_n) \leq d(s_m, L) + d(L, s_n)
\]
is also arbitrarily small. Then we have that $(s_n)_{n \in \N}$ is a Cauchy sequence, as desired.
\end{proof}
The key idea behind this proof is that the definitions of Cauchy sequence and convergent sequence differ primarily in that the latter explicitly references a limit while the former does not. In other words, we may think of a convergent sequence as one which ``gets close to something'' while a Cauchy sequence is one which ``gets close together.'' Our intuition, formalized by the triangle inequality, shows that ``getting close to something'' necessarily means ``getting close together,'' and this is our result.
Note that the converse of this statement is \emph{not} true. In fact, motivated by this theorem, we define a special class of metric spaces (the topologically complete spaces) as those in which the converse does hold. Certain large, important classes of metric spaces are complete (see e.g. \cite{rudin} Theorem 3.11).
\begin{thebibliography}{99}
\bibitem[R]{rudin} W. Rudin, \textit{Principles of Mathematical Analysis}, 3rd edition. McGraw-Hill, 1976.
\end{thebibliography}
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