Factors in Absolutely Convergent Product Converge to One/Proof 2

Proof
We have that $\ds \prod_{n \mathop = 1}^\infty \paren {1 + a_n}$ is absolutely convergent.

Let $b_n = \paren {1 + a_n}$

Then $\ds \prod_{n \mathop = 1}^\infty b_n$ is absolutely convergent.

From Absolutely Convergent Product is Convergent, $\ds \prod_{n \mathop = 1}^\infty b_n$ is convergent.

By Factors in Convergent Product Converge to One:


 * $b_n \to 1$

Thus:
 * $a_n \to 0$