Factors in Absolutely Convergent Product Converge to One/Proof 2

Proof
Because $\displaystyle \prod_{n \mathop = 1}^\infty \left({1 + a_n}\right)$ is absolutely convergent, the real product $\displaystyle \prod_{n \mathop = 1}^\infty (1+|a_n|)$ is convergent.

By Factors in Convergent Product Converge to One, $|a_n|\to0$.

Thus $a_n\to0$.