Factors in Absolutely Convergent Product Converge to One/Proof 1

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

Expanding out the product:


 * $1 + \paren {a_1 + a_2 + a_3 + \cdots} + \paren{a_1a_2 + a_1a_3 + \cdots} + \cdots$

Hence $\ds \sum_{n \mathop = 1}^\infty a_n$ is absolutely convergent.

By Terms in Convergent Series Converge to Zero:
 * $a_n \to 0$