Uniform Absolute Convergence of Infinite Product of Complex Functions

Theorem
Let $X$ be a compact topological space.

Let $\left\langle{f_n}\right\rangle$ be a sequence of continuous functions $X \to \C$.

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

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

Also see

 * Infinite Product of Analytic Functions