Absolutely Convergent Product is Convergent

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

Let $\mathbb K$ be complete.

Let the infinite product $\displaystyle \prod_{n \mathop = 1}^\infty \left({1 + a_n}\right)$ be absolutely convergent.

Then it is convergent or divergent to $0$.

Proof
Let $P_n$ and $Q_n$ denote the $n$th partial products of $\displaystyle \prod_{n \mathop = 1}^\infty \left({1 + a_n}\right)$ and $\displaystyle \prod_{n \mathop = 1}^\infty \left({1 + \left\lvert{a_n}\right\rvert}\right)$ respectively.

We show that $\left\langle{P_n}\right\rangle$ is Cauchy.

We have, for $m > n$:

Because $\left\langle{Q_n}\right\rangle$ converges, $\left\langle{Q_n}\right\rangle$ is Cauchy.

By the above inequality, $\left\langle{P_n}\right\rangle$ is Cauchy.

Because $\mathbb K$ is complete, $\left\langle{P_n}\right\rangle$ converges.

Also see

 * Absolute Convergence of Infinite Product of Complex Numbers