Absolutely Convergent Product Does not Diverge to Zero/Proof 2

Proof
We have that $\displaystyle \prod_{n \mathop = 1}^\infty \left({1 - |a_n|}\right)$ is absolutely convergent.

By Factors in Absolutely Convergent Product Converge to One, $|a_n|<1$ for $n\geq n_0$.

Thus $\displaystyle \sum_{n \mathop = n_0}^\infty \log \left({1 - |a_n|}\right)$ is absolutely convergent.

Suppose that the product diverges to $0$.

Then $\displaystyle \prod_{n=n_0}^\infty(1+a_n) = 0$.

By Norm of Limit, $\displaystyle \prod_{n=n_0}^\infty|1+a_n| = 0$.

By the Triangle Inequality and Squeeze Theorem, $\displaystyle \prod_{n=n_0}^\infty(1-|a_0|) = 0$.

By Logarithm of Infinite Product of Real Numbers, $\displaystyle \sum_{n \mathop = n_0}^\infty \log \left({1 - |a_n|}\right)$ diverges to $-\infty$.

This is a contradiction.