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

Convergence criteria

A sequence of functions from S to M is pointwise Cauchy if, for each x ∈ S, the sequence 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:
A sequence of functions from a set S to a metric space U is said to be uniformly Cauchy if: