Infinite product $\prod\limits_{n=1}^{\infty} (1-a_{n}) = 0$ implies divergence of $\sum_{n=1}^{\infty} a_{n}$ or not?

Question

If $\displaystyle\prod_{n=1}^{\infty} (1-a_{n}) = 0$ then is it always true that $\displaystyle\sum_{n=1}^{\infty} a_{n}$ diverges? ($0 \leq a_{n} < 1) $

Answer 1

If $a_n$ does not approach $0$ as $n \to \infty$, then $\displaystyle\sum_{n = 1}^{\infty}a_n$ diverges. So, assume $\displaystyle\lim_{n \to \infty}a_n = 0$

If $\displaystyle\prod_{n = 1}^{\infty}(1-a_n) = 0$, then $\displaystyle\sum_{n = 1}^{\infty}-\ln(1-a_n) = \infty$. Now, use the limit comparison test.