Uniformly Absolutely Convergent Product is Uniformly Convergent

Theorem
Let $X$ be a set.

Let $\struct {\mathbb K, \norm {\, \cdot \,} }$ be a valued field.

Let $\mathbb K$ be complete.

Let $\sequence {f_n}$ be a sequence of bounded mappings $f_n: X \to \mathbb K$.

Let the infinite product $\ds \prod_{n \mathop = 1}^\infty f_n$ converge uniformly absolutely on $X$.

Then it converges uniformly.