Absolute Value of Absolutely Convergent Product is Absolutely Convergent

Theorem
Let $\struct {\mathbb K, \size{\,\cdot\,}}$ be a valued field. Let the infinite product $\displaystyle \prod_{n \mathop = 1}^\infty a_n$ converge absolutely to $a\in\mathbb K$.

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

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

By the Triangle Inequality:
 * $\left\vert{\left\vert{a_n}\right\vert - 1}\right\vert \le \left\vert{a_n - 1}\right\vert$

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

By Absolute Value is Continuous, its limit is $\left\vert{a}\right\vert$.

Also see

 * Absolute Value of Infinite Product, for related results