Uniformly Convergent iff Difference Under Supremum Metric Vanishes

Theorem
Let $X$ be a set.

Let $\struct {Y, d}$ be a metric space.

Let $S$ be the set of all bounded mappings from $X$ to $Y$.

Let $\sequence {f_n}_{n \mathop \in \N}$ be a sequence in $S$.

Let $f \in S$.

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

Then:
 * $\sequence {f_n}$ converges uniformly to $f$ on $S$


 * $\map {d_S} {f_n, f} \to 0$ as $n \to \infty$.
 * $\map {d_S} {f_n, f} \to 0$ as $n \to \infty$.