Uniform Absolute Convergence of Infinite Product of Complex Functions

Theorem
Let $X$ be a compact metric space.

Let $(f_n)$ be a sequence of functions $X\to\C$.

Suppose $\displaystyle\sum_{n=1}^\infty f_n$ converges uniformly absolutely on $X$.

Then $f(x)=\displaystyle\prod_{n=1}^\infty(1+f_n(x))$ converges uniformly absolutely on $X$, $f$ is continuous and there exists $n_0\in\N$ such that $\displaystyle\prod_{n=n_0}^\infty(1+f_n(x))$ has no zeroes.