Product of Convergent and Divergent Product is Divergent

Theorem
Let $\struct {\mathbb K, \norm{\,\cdot\,}}$ be a valued field.

Let $\displaystyle \prod_{n=1}^\infty a_n$ be convergent.

Let $\displaystyle \prod_{n=1}^\infty b_n$ be divergent.

Then $\displaystyle \prod_{n=1}^\infty a_nb_n$ is divergent.

Also see

 * Product of Convergent Products is Convergent
 * Product of Absolutely Convergent Products is Absolutely Convergent