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.

A metric space in which every Cauchy sequence converges is called complete. The real numbers are complete, and the standard construction of the real numbers involves Cauchy sequences of rational numbers.

Cauchy sequences are named after the French mathematician Augustin Louis Cauchy.