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 Subspace of Euclidean Space is Bounded

Uniformly Convergent Sequence Multiplied with Function

$\blacksquare$