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.


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.