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 $\ds \prod_{n \mathop = 1}^\infty a_n$ converge absolutely to $a \in \mathbb K$.


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


Proof

By absolute convergence of $\ds \prod_{n \mathop = 1}^\infty a_n$, $\ds \sum_{n \mathop = 1}^\infty \paren {a_n - 1}$ converges absolutely.

By the Triangle Inequality:

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

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

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

$\blacksquare$


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