Proof that the Limit of a Uniformly Convergent Sequence of Bounded Mappings is Bounded

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Theorem

The limit of a uniformly convergent sequence of bounded mappings is bounded.

Proof

Letwhereis an arbitrary set andis a metric space. The sequenceconverges uniformly toif and only if

Consider the spaceof all bounded mappingswith the metric defined by

We have

Hence in the spacethe conditionmeans that the sequence converges uniformly to

Letand letwhereis a uniformly convergent sequence of bounded mappings. Choosesuch that

We have

Henceis bounded.