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For functions into a complete metric space, uniform convergence is equivalent to being uniformly Cauchy.
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 complete metric space is a metric space in which every Cauchy sequence converges (as a convergent sequence) to a point of .
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,
A sequence of functions into a metric space is uniform Cauchy on if for every there exists such that for all and all ,
Equivalently,
The uniform metric on a set of bounded real-valued functions on is defined by
Equivalently, where is the supremum norm.