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\author{C. Desjardins and J. Lewis}
\title{Cauchy Sequences and Convergence}
\date{October 19, 2009}


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\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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