Proof that the Limit of a Uniformly Convergent Sequence of Bounded Mappings is Bounded
The limit of a uniformly convergent sequence of bounded mappings is bounded.
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
Hence in the spacethe conditionmeans that the sequence converges uniformly to
Letand letwhereis a uniformly convergent sequence of bounded mappings. Choosesuch that