Uniformly Cauchy sequence

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

  • For all \varepsilon > 0, there exists N>0 such that for all x\in S: d(f_{n}(x), f_{m}(x)) < \varepsilon whenever m, n > N.

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[edit]

A sequence of functions {fn} from S to M is pointwise Cauchy if, for each xS, the sequence {fn(x)} is a Cauchy sequence in M. This is a weaker condition than being uniformly Cauchy. 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 fn : SM tends uniformly to a unique continuous function f : SM.

Generalization to uniform spaces[edit]

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

  • For all x\in S and for any entourage \varepsilon, there exists N>0 such that d(f_{n}(x), f_{m}(x)) < \varepsilon whenever m, n > N.

See also[edit]