Uniformly Convergent Sequence Multiplied with Function/Corollary

Corollary to Uniformly Convergent Sequence Multiplied with Function
Let $X$ be a compact topological space. 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 continuous.

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

Proof
Follows directly from:
 * Continuous Function on Compact Space is Bounded
 * Uniformly Convergent Sequence Multiplied with Function