Uniform Absolute Convergence of Infinite Product of Complex Functions

Theorem
Let $X$ be a compact metric space.

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

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

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