Uniform Convergence is Hereditary

Theorem
Let $X$ be a metric space.

Let $\left\langle{ f_n }\right\rangle$ be a sequence of mappings defined on $X$.

Let $\left\langle{ f_n }\right\rangle$ be uniformly convergent on $S \subseteq X$.

Then $\left\langle{ f_n }\right\rangle$ is uniformly convergent on every metric subspace of $S$.

That is, uniform convergence is a hereditary property of a metric space.