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 $\varepsilon > 0$ 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.