Product of Uniformly Convergent Sequences of Bounded Functions is Uniformly Convergent

Theorem
Let $X = \struct {A, d}$ and $Y = \struct {B, \rho}$ be metric spaces.

Let $\sequence {f_n}$ and $\sequence {g_n}$ be sequences of mappings from $X$ to $Y$.

Let $\sequence {f_n}$ and $\sequence {g_n}$ be uniformly convergent on some subspace $S$ of $X$.

$\forall n \in \N$, let $f_n$ and $g_n$ be bounded.

Then the sequence $\sequence {f_n g_n}$ is uniformly convergent on $S$.