Uniform Convergence is Hereditary
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Theorem
Let $M = \struct {A, d}$ be a metric space.
Let $\sequence {f_n}$ be a sequence of mappings defined on $A$.
Let $\sequence {f_n}$ be uniformly convergent on $S \subseteq A$.
Then $\sequence {f_n}$ is uniformly convergent on every metric subspace of $S$.
That is, uniform convergence is a hereditary property of a metric space.
Proof
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