Uniformly Convergent iff Difference Under Supremum Metric Vanishes

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

Let $\sequence {f_n}$ 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 $\sequence {f_n}$ converges uniformly to $f$ on $S$ $\map {d_S} {f_n, f} \to 0$ as $n \to \infty$.