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 $\left\langle{f_n}\right\rangle$ be a sequence of bounded mappings $f_n: X \to \mathbb K$.

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

Then it converges uniformly.