# Uniformly Cauchy sequence

In mathematics, a sequence of functions from a set *S* to a metric space *M* is said to be **uniformly Cauchy** if:

- For all , there exists such that for all : whenever .

Another way of saying this is that as , where the uniform distance between two functions is defined by

## Convergence criteria

A sequence of functions {*f*_{n}} from *S* to *M* is **pointwise** Cauchy if, for each *x* ∈ *S*, the sequence {*f*_{n}(*x*)} is a Cauchy sequence in *M*. This is a weaker condition than being uniformly Cauchy.

In general a sequence can be pointwise Cauchy and not pointwise convergent, or it can be uniformly Cauchy and not uniformly convergent. Nevertheless, if the metric space *M* is complete, then any pointwise Cauchy sequence converges pointwise to a function from *S* to *M*. Similarly, any uniformly Cauchy sequence will tend uniformly to such a function.

The uniform Cauchy property is frequently used when the *S* is not just a set, but a topological space, and *M* is a complete metric space. The following theorem holds:

- Let
*S*be a topological space and*M*a complete metric space. Then any uniformly Cauchy sequence of continuous functions*f*_{n}:*S*→*M*tends uniformly to a unique continuous function*f*:*S*→*M*.

## Generalization to uniform spaces

A sequence of functions from a set *S* to a metric space *U* is said to be **uniformly Cauchy** if:

- For all and for any entourage , there exists such that whenever .