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 $(f_n)$ be a sequence of mappings $f_n : X\to V$.

Let $(f_n)$ be uniformly convergent. Let $g :X\to \mathbb K$ be continuous.

Then $(f_ng)$ is uniformly convergent.

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