Absolute Value of Absolutely Convergent Product is Absolutely Convergent

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Theorem

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

Let the infinite product $\displaystyle \prod_{n \mathop = 1}^\infty a_n$ converge absolutely to $a\in\mathbb K$.


Then $\displaystyle \prod_{n \mathop = 1}^\infty \norm{a_n}$ converges absolutely to $\norm{a}$.


Proof

By absolute convergence of $\displaystyle \prod_{n \mathop = 1}^\infty a_n$, $\displaystyle \sum_{n \mathop = 1}^\infty \left({a_n - 1}\right)$ converges absolutely.

By the Triangle Inequality:

$\size{\norm{a_n} - 1} \le \norm{a_n - 1}$

By the Comparison Test, $\displaystyle \prod_{n \mathop = 1}^\infty \norm{a_n}$ converges absolutely.

By Absolute Value is Continuous, its limit is $\norm{a}$.

$\blacksquare$


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