Definition:Uniform Convergence of Product/Compact Space

Definition
Let $X$ be a compact topological space.

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

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

The infinite product $\displaystyle \prod_{n \mathop = 1}^\infty f_n$ converges uniformly there exists $n_0 \in \N$ such that the sequence of partial products of $\displaystyle \prod_{n \mathop = n_0}^\infty f_n$ converges uniformly and is nonzero.