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.