Uniformly Convergent Sequence of Bounded Functions is Uniformly Bounded

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

Let $\left \langle{f_i}\right \rangle_{i \in I}$ be a uniformly convergent sequence of mappings $f_i: X \to Y$.

$\forall i \in I$, let $f_i$ be bounded.

Then $\left \langle{f_i}\right \rangle$ is uniformly bounded.