Definition:Uniform Absolute Convergence of Product/Complex Functions

Definition
Let $X$ be a set.

Let $\left \langle {f_n} \right \rangle$ be a sequence of mappings $f_n: S \to \C$.

The infinite product $\displaystyle \prod_{n \mathop = 1}^\infty \left({1 + f_n}\right)$ converges uniformly absolutely there exists $n_0\in\N$ such that:
 * $f_n(x) \ne -1$ for $n \geq n_0$ and $x\in X$.
 * The series $\displaystyle \sum_{n \mathop = n_0}^\infty \log \left({1 + f_n}\right)$ is uniformly absolutely convergent.