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On a compact metric space, equicontinuity allows pointwise Cauchy behavior on a dense set to upgrade to uniform Cauchy behavior.
Equicontinuity and dense sets lemma. Let be a compact metric space, let be dense, and let be an equicontinuous sequence of functions . If is Cauchy for every , then is uniformly Cauchy on .
Because is complete, the uniform Cauchy criterion implies that converges uniformly on .
A dense set in a topological space is a subset such that its closure is all of , i.e. .
A family of functions from a metric space to a metric space is equicontinuous at a point if for every there exists such that for every and every ,
Equicontinuity strengthens the statement that each is a continuous map: here the same must work simultaneously for all functions in the family. A family that is equicontinuous at every point is an equicontinuous family.
A sequence of functions into a metric space is uniform Cauchy on if, for every , there exists such that for all and ,
Equivalently,
Uniform Cauchy criterion: Let be a set and let be a complete metric space. A sequence of functions converges uniformly on if and only if it is uniformly Cauchy, meaning: for every there exists such that for all and all ,
Equivalently, uniform convergence is exactly convergence in the uniform metric (and, for bounded real-valued functions, in the supremum norm).
A sequence of functions from a set into a metric space converges uniformly to a function if for every there exists such that for all and all ,
Equivalently,