Product of Absolutely Convergent Products is Absolutely Convergent

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

Let $\displaystyle \prod_{n=1}^\infty a_n$ converge absolutely.

Let $\displaystyle \prod_{n=1}^\infty b_n$ converge absolutely.

Then $\displaystyle \prod_{n=1}^\infty a_nb_n$ converges absolutely.

Proof
We have $a_nb_n-1 = (a_n-1)(b_n-1) + (a_n-1) + (b_n-1)$.

By the Triangle Inequality, $|a_nb_n-1| \leq |a_n-1||b_n-1| + |a_n-1| + |b_n-1|$.

By the absolute convergence, $\displaystyle \sum_{n=1}^\infty |a_n-1|$ and $\displaystyle \sum_{n=1}^\infty |b_n-1|$ converge.

By Inner Product of Absolutely Convergent Series, $\displaystyle \sum_{n=1}^\infty |a_n-1||b_n-1|$ converges.

By the Comparison Test, $\displaystyle \sum_{n=1}^\infty |a_nb_n-1|$ converges.

Also see

 * Product of Convergent and Divergent Product is Divergent
 * Product of Convergent Products is Convergent