Absolute Value of Infinite Product
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Theorem
Let $\struct {\mathbb K, \norm{\,\cdot\,}}$ be a valued field.
Let $\sequence{a_n}$ be a sequence in $\mathbb K$.
Convergence
Let the infinite product $\ds \prod_{n \mathop = 1}^\infty a_n$ converge to $a \in \mathbb K$.
Then $\ds \prod_{n \mathop = 1}^\infty \norm {a_n}$ converges to $\norm{a}$.
Absolute Convergence
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 $\norm a$.
Divergence to zero
The infinite product $\ds \prod_{n \mathop = 1}^\infty a_n$ diverges to $0$ if and only if $\ds \prod_{n \mathop = 1}^\infty \norm {a_n}$ diverges to $0$.