Absolutely Convergent Product Does not Diverge to Zero/Proof 2

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Theorem

Let $\struct {\mathbb K, \norm {\, \cdot \,} }$ be a valued field.

Let the infinite product $\displaystyle \prod_{n \mathop = 1}^\infty \paren {1 + a_n}$ be absolutely convergent.


Then it is not divergent to $0$.


Proof

We have that $\displaystyle \prod_{n \mathop = 1}^\infty \paren {1 - \norm {a_n} }$ is absolutely convergent.

By Factors in Absolutely Convergent Product Converge to One, $\norm {a_n} < 1$ for $n \ge n_0$.

Thus $\displaystyle \sum_{n \mathop = n_0}^\infty \map \ln {1 - \norm {a_n} }$ is absolutely convergent.

Aiming for a contradiction, suppose the product diverges to $0$.

Then:

$\displaystyle \prod_{n \mathop = n_0}^\infty \paren {1 + a_n} = 0$

By Norm of Limit:

$\displaystyle \prod_{n \mathop = n_0}^\infty \norm {1 + a_n} = 0$

By the Triangle Inequality and Squeeze Theorem:

$\displaystyle \prod_{n \mathop = n_0}^\infty \paren {1 - \norm {a_0} } = 0$

By Logarithm of Infinite Product of Real Numbers:

$\displaystyle \sum_{n \mathop = n_0}^\infty \map \ln {1 - \norm {a_n} }$

diverges to $-\infty$.

This is a contradiction.

$\blacksquare$