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 and a distance function $d$ , a sequence $(x_{n})_{n = 0}^{\infty}$ is Cauchy if for every $ε > 0$ there exists an $N$ such that for all $m, n > N$ , we have that $d (x_{m}, x_{n}) < ε$ .

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.