Absolutely Convergent Product is Convergent
Theorem
Let $\struct {\mathbb K, \norm {\,\cdot\,} }$ be a valued field.
Let $\mathbb K$ be complete.
Let the infinite product $\ds \prod_{n \mathop = 1}^\infty \paren {1 + a_n}$ be absolutely convergent.
Then it is convergent.
Proof
Let $P_n$ and $Q_n$ denote the $n$th partial products of $\ds \prod_{n \mathop = 1}^\infty \paren {1 + a_n}$ and $\ds \prod_{n \mathop = 1}^\infty \paren {1 + \norm {a_n} }$ respectively.
We show that $\sequence {P_n}$ is Cauchy.
We have, for $m > n$:
\(\ds \norm {P_m - P_n}\) | \(=\) | \(\ds \prod_{k \mathop = 1}^n \norm {1 + a_k} \cdot \norm {\prod_{k \mathop = n + 1}^m \paren {1 + a_k} - 1}\) | ||||||||||||
\(\ds \) | \(\) | \(\ds \) | ||||||||||||
\(\ds \) | \(\le\) | \(\ds \prod_{k \mathop = 1}^n \paren {1 + \norm {a_k} } \paren {\prod_{k \mathop = n + 1}^m \paren {1 + \norm {a_k} } - 1}\) | ||||||||||||
\(\ds \) | \(\) | \(\ds \) | ||||||||||||
\(\ds \) | \(=\) | \(\ds Q_m - Q_n\) |
Because $\sequence {Q_n}$ converges, $\sequence {Q_n}$ is Cauchy.
By the above inequality, $\sequence {P_n}$ is Cauchy.
Because $\mathbb K$ is complete, $\sequence {P_n}$ converges to some $a \in \mathbb K$.
By Absolutely Convergent Product Does not Diverge to Zero, the product converges.
$\blacksquare$