Product of Convergent and Divergent Product is Divergent

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

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

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

Then $\displaystyle \prod_{n \mathop = 1}^\infty a_n b_n$ is divergent.

Also see

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