Absolutely Convergent Product is Convergent

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

Suppose $\mathbb K$ is complete.

Suppose that the infinite product $\displaystyle\prod_{n=1}^\infty(1+a_n)$ is absolutely convergent.

Then it is convergent or divergent to $0$.

Proof
Let $P_n$ and $Q_n$ denote the $n$th partial products of $\prod_{n=1}^\infty(1+a_n)$ and $\prod_{n=1}^\infty(1+|a_n|)$ respectively.

We show that $(P_n)$ is Cauchy.

We have, for $m>n$,

Because $(Q_n)$ converges, $(Q_n)$ is Cauchy.

By the above inequality, $(P_n)$ is Cauchy.

Because $\mathbb K$ is complete, $(P_n)$ converges.

Also see

 * Absolute Convergence of Infinite Product of Complex Numbers