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 $\left\vert{a}\right\vert$.


 * $\displaystyle \prod_{n \mathop = 1}^\infty a_n$ diverges to $0$  $\displaystyle \prod_{n \mathop = 1}^\infty \left\vert{a_n}\right\vert$ diverges to $0$.

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

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

Let $P_n \to a$.

Then by Convergence of Absolute Value of Sequence:
 * $\left\vert{P_n}\right\vert \to \left\vert{a}\right\vert$

From Convergence of Absolute Value of Sequence:
 * $P_n \to 0$ $\left\vert{P_n}\right\vert \to 0$.