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 $\sequence {a_n}$ be a sequence in $\mathbb K$.
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 definition of 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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Also see
- Absolute Value of Infinite Product, for related results