Uniform Convergence is Hereditary

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.