Absolutely Convergent Product Does not Diverge to Zero/Proof 1
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Theorem
Let $\struct {\mathbb K, \norm {\, \cdot \,} }$ be a valued field.
Let the infinite product $\ds \prod_{n \mathop = 1}^\infty \paren {1 + a_n}$ be absolutely convergent.
Then it is not divergent to $0$.
Proof
By Factors in Absolutely Convergent Product Converge to One, $\norm {a_n} < 1$ for $n \ge n_0$.
Let $n_1 \ge n_0$.
Aiming for a contradiction, suppose the product diverges to $0$.
Then:
- $\ds \prod_{n \mathop = n_1}^\infty \paren {1 + a_n} = 0$
By Norm of Limit:
- $\ds \prod_{n \mathop = n_1}^\infty \norm {1 + a_n} = 0$
By the Triangle Inequality and Squeeze Theorem:
- $\ds \prod_{n \mathop = n_1}^\infty \paren {1 - \norm {a_n} } = 0$
By the Weierstrass Product Inequality, we have for $N \ge n_1$:
- $\ds \prod_{n \mathop = n_1}^N \paren {1 - \norm {a_n} } \ge 1 - \sum_{n \mathop = n_1}^N \norm{a_n}$
Taking limits:
- $0 \ge 1 - \ds \sum_{n \mathop = n_1}^\infty \norm {a_n}$
Because $\ds \prod_{n \mathop = 1}^\infty \paren {1 + a_n}$ is absolutely convergent:
- $\ds \sum_{n \mathop = n_1}^\infty \norm {a_n} < 1$
for $n_1$ sufficiently large.
This is a contradiction.
$\blacksquare$