A **Cauchy sequence** is a sequence in which as the sequence progresses, all the terms get closer and closer together. It is closely related to the idea of a [-convergent_sequence].

## Definition

In any [-metric_space] with a set $~$X$~$ and a distance function $~$d$~$, a sequence $~$(x_n)_{n=0}^\infty$~$ is Cauchy if for every $~$\varepsilon > 0$~$ there exists an $~$N$~$ such that for all $~$m, n > N$~$, we have that $~$d(x_m, x_n) < \varepsilon$~$.

In the real numbers, the distance between two numbers is usually expressed as their difference, or $~$|x_m - x_n|$~$.

## Complete metric space

In a [ complete metric space], every Cauchy sequence is convergent. In particular, the real numbers are a complete metric space.