Uniformly Convergent Product Satisfies Uniform Cauchy Criterion

Theorem
Let $X$ be a compact topological space.

Let $\struct {\mathbb K, \norm{\,\cdot\,}}$ 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 on $X$.

Then $\displaystyle \prod_{n \mathop = 1}^\infty f_n$ satisfies the uniform Cauchy condition for products.

Also see

 * Tail of Uniformly Convergent Product Converges Uniformly to One
 * Factors in Uniformly Convergent Product Converge Uniformly to One