Cauchy sequence

A sequence x₁, x₂, x₃, ... in a metric space (M, d) is called a Cauchy sequence (or Cauchy for short) if 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 have a limit in M. Nonetheless, this does not need to be the case.

A metric space in which every Cauchy sequence has a limit is called complete. The real numbers are complete, and the standard construction of the real numbers involves Cauchy sequences of rational numbers. The rational numbers themselves are not complete. See completeness for an example of a Cauchy sequences of rational numbers that does not have a rational limit.

Every convergent sequence is a Cauchy sequence, and every Cauchy sequence is bounded. If f : M -> N is a uniformly continuous map between the metric spaces M and N and (x_n) is a Cauchy sequence in M , then (f(x_n)) is a Cauchy sequence in N. If (x_n) and (y_n) are two Cauchy sequences in the rational, real or complex numbers, then the sum (x_n + y_n) and the product (x_ny_n) are also Cauchy sequences.

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