Complete metric space

A sequence x₁, x₂, ... in a metric space (X, d) is said to be a Cauchy sequence (or is said to be Cauchy) if, for every ε>0, there is a natural number N such that for all m,n > N,

d(x_n, x_m) < ε

That is, after a certain pont, terms of the sequence are "close".

The metric space (X, d) is said to be complete if every Cauchy sequence of points in X converges (see limit) towards a point in X.

This is an important concept for many branches of mathematics. The real numbers with any of the standard metrics are complete.