Absolutely Convergent Product Does not Diverge to Zero/Proof 2

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

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

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

the product diverges to $0$.

Then:
 * $\ds \prod_{n \mathop = n_0}^\infty \paren {1 + a_n} = 0$

By Norm of Limit:
 * $\ds \prod_{n \mathop = n_0}^\infty \norm {1 + a_n} = 0$

By the Triangle Inequality and Squeeze Theorem:
 * $\ds \prod_{n \mathop = n_0}^\infty \paren {1 - \norm {a_0} } = 0$

By Logarithm of Infinite Product of Real Numbers:
 * $\ds \sum_{n \mathop = n_0}^\infty \map \ln {1 - \norm {a_n} }$

diverges to $-\infty$.

This is a contradiction.