Factors in Absolutely Convergent Product Converge to One

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Theorem

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

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


Then:

$a_n \to 0$


Proof 1

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

By Terms in Convergent Series Converge to Zero, $a_n\to0$.

$\blacksquare$


Proof 2

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

By Factors in Convergent Product Converge to One, $\norm{a_n} \to 0$.

Thus $a_n\to0$.

$\blacksquare$


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