Uniform Product of Continuous Functions is Continuous

Theorem
Let $X$ be a metric space.

Let $\mathbb K$ be a field with absolute value $\left\vert{\, \cdot \,}\right\vert$.

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

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

Then $f$ is continuous.

Also see

 * Infinite Product of Analytic Functions is Analytic