Absolute Value of Infinite Product

Theorem
Let $\mathbb K$ be a field with absolute value $\left\vert{\cdot}\right\vert$.

Let $\displaystyle \prod_{n \mathop = 1}^\infty a_n$ be an infinite product.

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

$\displaystyle \prod_{n \mathop = 1}^\infty a_n$ diverges to $0$ iff $\displaystyle \prod_{n \mathop = 1}^\infty |a_n|$ diverges to $0$.

Proof
Let $P_n$ denote the $n$th partial product of $\displaystyle \prod_{n \mathop = 1}^\infty a_n$.

Then $|P_n|$ is the $n$th partial product of $\displaystyle \prod_{n \mathop = 1}^\infty a_n$.

If $P_n\to a$, then by Convergence of Absolute Value of Sequence, $|P_n|\to|a|$.

From Convergence of Absolute Value of Sequence, $P_n\to0$ iff $|P_n|\to0$.