Product of Uniformly Convergent Sequences of Bounded Functions is Uniformly Convergent

Theorem
Let $X = \left({A, d}\right)$ and $Y = \left({B, \rho}\right)$ be metric spaces.

Let $\left \langle{f_n}\right \rangle$ and $\left \langle{g_n}\right \rangle$ be sequences of mappings from $X$ to $Y$.

Let $\left \langle{f_n}\right \rangle$ and $\left \langle{g_n}\right \rangle$ be uniformly convergent on some subspace $S$ of $X$.

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

Then the sequence $\left \langle{f_n g_n}\right \rangle$ is uniformly convergent on $S$.