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.

Formally, a sequence *x*_{1}, *x*_{2}, *x*_{3}, ... 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: a sequence of rational numbers can have the square root of two as its limit, for example. See Complete space for an example of a Cauchy sequence 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*_{n}*y*_{n}) are also Cauchy sequences.

## Generalisation

A net (*x*_{α}) in a uniform space *X* is a *Cauchy net* if for every entourage *V* there exists an α_{0} such that for all α, β > α_{0} we have (*x*_{α}, *x*_{β}) in *V*.