Uniformly Convergent iff Difference Under Supremum Metric Vanishes

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

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

Let $\left\Vert{ \cdot }\right\Vert_S$ denote the supremum norm on $S \subseteq X$.

Then $\left\langle{ f_n }\right\rangle$ converges uniformly to $f$ on $S$  $\left\Vert{ f_n - f }\right\Vert_S \to 0$.