Absolutely Convergent Product Does not Diverge to Zero/Proof 1

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

Let $n_1 \ge n_0$.

the product diverges to $0$.

Then:
 * $\displaystyle \prod_{n \mathop = n_1}^\infty \paren {1 + a_n} = 0$

By Norm of Limit:
 * $\displaystyle \prod_{n \mathop = n_1}^\infty \norm {1 + a_n} = 0$

By the Triangle Inequality and Squeeze Theorem:
 * $\displaystyle \prod_{n \mathop = n_1}^\infty \paren {1 - \norm {a_n} } = 0$

By the Weierstrass Product Inequality, we have for $N \ge n_1$:
 * $\displaystyle \prod_{n \mathop = n_1}^N \paren {1 - \norm {a_n} } \ge 1 - \sum_{n \mathop = n_1}^N \norm{a_n}$

Taking limits:
 * $0 \ge 1 - \displaystyle \sum_{n \mathop = n_1}^\infty \norm {a_n}$

Because $\displaystyle \prod_{n \mathop = 1}^\infty \paren {1 + a_n}$ is absolutely convergent:
 * $\displaystyle \sum_{n \mathop = n_1}^\infty \norm {a_n} < 1$

for $n_1$ sufficiently large.

This is a contradiction.