Factors in Absolutely Convergent Product Converge to One/Proof 2

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:

$a_n \to 0$

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:

$a_n \to 0$

$\blacksquare$