Cauchy sequence

A Cauchy sequence is a sequence x₁, x₂, x₃, ... in a metric space with the property that for every positive real number r, there is an integer N such that for all integers m and n greater than N the distance d(x_m, x_n) is less than r. Roughly speaking, the terms of the sequence are getting closer and closer together in a way that suggests that the sequence ought to converge. Nonetheless, Cauchy sequences do not always converge.

Cauchy sequences are used to define the concept of completeness in metric spaces, and are also used in the usual set-theoretic construction of the real numbers. They are named after the French mathematician Augustin Louis Cauchy.