Absolute Value of Convergent Infinite Product

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

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

Proof
By Absoltue Value of Limit of Sequence, $\displaystyle \prod_{n \mathop = 1}^\infty |a_n| = |a|$.

It remains to show that the product converges.

By the convergence, there exists $n_0\in\N$ such that $a_n\neq0$ for $n\geq n_0$.

Then $|a_n|\neq0$ for $n\geq n_0$.

Let $P_n$ denote the $n$th partial product of $\displaystyle \prod_{n \mathop = n_0}^\infty a_n$.

Then $\left\vert{P_n}\right\vert$ is the $n$th partial product of $\displaystyle \prod_{n \mathop = n_0}^\infty |a_n|$.

Let $P_n$ converge to $b\in\mathbb K\setminus\{0\}$.

By Convergence of Absolute Value of Sequence:
 * $\left\vert{P_n}\right\vert \to \left\vert{b}\right\vert \in\mathbb K\setminus\{0\}$.

Thus $\displaystyle \prod_{n \mathop = 1}^\infty |a_n|$ converges.

Also see

 * Absolute Value of Infinite Product, for related results