Uniformly Convergent Sequence Multiplied with Function

Theorem
Let $X$ be a set.

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 bounded.

Then $(f_ng)$ is uniformly convergent.