Uniform Product of Continuous Functions is Continuous

Definition
Let $X$ be a compact 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 continuous mappings $f_n:X\to \mathbb K$.

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

Then the product is continuous.

Proof
Follows directly from:
 * Partial Products of Uniformly Convergent Product Converge Uniformly
 * Uniform Limit Theorem

Also see

 * Infinite Product of Analytic Functions