Uniform Product of Continuous Functions is Continuous

Theorem
Let $X$ be a metric space.

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

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

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

Then $f$ is continuous.

Also see

 * Infinite Product of Analytic Functions is Analytic