Factors in Absolutely Convergent Product Converge to One

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

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

Then $a_n\to0$.

Also see

 * Factors in Convergent Product Converge to One
 * Absolutely Convergent Product is Convergent (which relies on this result)