Equivalence of Definitions of Uniform Absolute Convergence of Product of Complex Functions

Definition
Let $X$ be a set.

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

Then the following definitions of absolute convergence of a product are equivalent:

2 implies 3
By Terms in Uniformly Convergent Series Converge Uniformly to Zero, there exists $n_0\in\N$ such that $|f_n(x)|\leq\frac12$ for $n\geq n_0$.

Then $f_n(x)\neq-1$ for all $n\geq n_0$ and $x\in X$.

By Bounds for Complex Logarithm:
 * $|\log(1 + f_n(x))|\leq \frac32 |f_n(x)|$

for $n\geq n_0$.

By Comparison Test for Uniformly Convergent Series,
 * $\displaystyle \sum_{n \mathop = n_0}^\infty \log(1+f_n)$

converge uniformly absolutely.