In mathematical analysis, a Cauchy sequence is a sequence whose terms become arbitrarily close to each other as the sequence progresses. They are named after the French mathematician Augustin Louis Cauchy. They are of interest because, given certain conditions, all such sequences converge to a limit, and one can test for "Cauchiness" without having the value of the actual limit.

Formally, a Cauchy sequence is a sequence

in a metric space (M, d) such that for every positive real number r > 0, there is an integer N such that for all integers m,n > 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 may not be the case.

A metric space X in which every Cauchy sequence has a limit (in X) 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: for example the following sequence of rational numbers has the irrational square root of two as its limit (each numerator is the square of the previous numerator plus twice the square of the previous denominator, while each denominator is twice the product of the previous numerator and denominator).

Every convergent sequence is a Cauchy sequence, and every Cauchy sequence is bounded. If 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.

There is also a concept of Cauchy sequence for a topological vector space X: Pick a local base B for X about 0; then (x_k) is a Cauchy sequence if for all members V of B, there is some number N such that whenever n,m > N, x_n - x_n is an element of V. If the topology of X is compatible with a translation-invariant metric d, the two definitions agree.