Absolutely Convergent Product Does not Diverge to Zero

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

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

Then it is not divergent to $0$.

Also see

 * Absolutely Convergent Product is Convergent