Tail of Uniformly Convergent Product Converges Uniformly to One

Theorem
Let $X$ be a compact topological space.

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

Let $\sequence {f_n}$ 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 for all $N \in \N$, $\displaystyle \prod_{n \mathop = N}^\infty f_n$ converges uniformly and the sequence $\displaystyle \prod_{n \mathop = N}^\infty f_n$ converges uniformly to $1$.

Also see

 * Factors in Uniformly Convergent Product Converge Uniformly to One