Uniformly Convergent iff Difference Under Supremum Metric Vanishes

Theorem
Let $X$ and $Y$ be metric spaces

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

Let $f: X \to Y$ be a mapping.

Let $d_S: S \times S \to Y$ denote the supremum metric on $S \subseteq X$.

Then $\left\langle{ f_n }\right\rangle$ converges uniformly to $f$ on $S$  $d_S \left({ f_n, f }\right) \to 0$ as $n \to \infty$.