# Cauchy sequence

A Cauchy sequence is defined to be a sequence $x_n$ such that, for any value $\varepsilon$, we have $|x_n - x_m|<\varepsilon$ for all sufficiently large $m, n$.

In a complete metric space, all Cauchy sequences have a limit (in fact, this is the definition of "complete"). In a metric space $M$ that is not complete, we can construct another metric space $\overline{M}$ (called the "completion" of that metric space) with the entries being the Cauchy sequences in $M$ under the equivalence relation that two sequences $x_n$ and $y_n$ are equivalent if $\lim_{n\to\infty} |x_n - y_n| = 0$, and with the metric $|\{x_n\}_{n=1}^{\infty}, \{y_n\}_{n=1}^{\infty}| = \lim_{n\to\infty} |x_n - y_n|$; the completion of a metric space is always complete.

This fact is useful because we can use it to give an unambiguous construction of the real numbers as the completion of the rational numbers under the absolute value metric. It is also useful to give formal proofs of convergence for certain sequences.