Uniformly Convergent Sequence Multiplied with Function

Theorem
Let $X$ be a set.

Let $V$ be a normed vector space over $\mathbb K$.

Let $\left\langle{f_n}\right\rangle$ be a sequence of mappings $f_n: X \to V$.

Let $\left\langle{f_n}\right\rangle$ be uniformly convergent.

Let $g: X \to \mathbb K$ be bounded.

Then $\left\langle{f_n g}\right\rangle$ is uniformly convergent.

Also see

 * Dirichlet's Test for Uniform Convergence