Uniformly Cauchy sequence

In mathematics, a sequence of functions \{f_{n}\} from a set S to a metric space M is said to be uniformly Cauchy if:

Another way of saying this is that d_u (f_{n}, f_{m}) \to 0 as m, n \to \infty, where the uniform distance d_u between two functions is defined by

d_{u} (f, g) := \sup_{x \in S} d (f(x), g(x)).

Convergence criteria

A sequence of functions {fn} from S to M is pointwise Cauchy if, for each x S, the sequence {fn(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:

Generalization to uniform spaces

A sequence of functions \{f_{n}\} from a set S to a metric space U is said to be uniformly Cauchy if:

See also


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