Absolute Value of Absolutely Convergent Product is Absolutely Convergent

Theorem
Let $\mathbb K$ be a field with absolute value $\left\vert{\, \cdot \,}\right\vert$. Let the infinite product $\displaystyle \prod_{n \mathop = 1}^\infty a_n$ converge absolutely to $a\in\mathbb K$.

Then $\displaystyle \prod_{n \mathop = 1}^\infty |a_n|$ converges absolutely to $|a|$.

Proof
By absolute convergence of $\displaystyle \prod_{n \mathop = 1}^\infty a_n$, $\displaystyle \sum_{n \mathop = 1}^\infty(a_n-1)$ converges absolutely.

By the Triangle Inequality, $|a_n|-1\leq |a_n-1|$.

By the Comparison Test, $\displaystyle \prod_{n \mathop = 1}^\infty |a_n|$ converges absolutely.

By Absolute Value is Continuous, its limit is $|a|$.

Also see

 * Absolute Value of Infinite Product, for related results