Cauchy sequence

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 $\epsilon >0$ there exists an $N$ such that for all $m,n>N$, we have that $d(x_m,x_n)<\epsilon$.

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.