A **Cauchy sequence** is a sequence *x*_{1}, *x*_{2}, *x*_{3}, ... 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.