Absolute Value of Uniformly Convergent Product

Theorem

Let $X$ be a compact topological space.

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

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

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

Then $\displaystyle \prod_{n \mathop = 1}^\infty \left\vert{f_n}\right\vert$ converges uniformly to $\left\vert{f}\right\vert$.