Factors in Absolutely Convergent Product Converge to One

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

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

Then:
 * $a_n \to 0$

Also see

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