Assume that for all $n \in \mathbb N$ we have that $p_n = 1- s_n$ , where $s_n \in (0 ,1)$. Prove that $\prod _{n=0}^ \infty p_n$ converges non-zero number iff $\sum_{n=0}^\infty s_n$ converges.
My Attempt: I can show that if $\sum_{n=0}^\infty s_n$ converges then $\prod _{n=0}^ \infty p_n$ converges by using the inequality $e^{-x} \geq 1-x$. But I am having difficulty to show that the limit is non-zero and the converse part is true. Can anyone please help me ?
Note that $$\prod_{n=1}^Np_n=\exp\left(\ln\left(\prod_{n=1}^Np_n\right)\right)=\exp\left(\sum_{n=1}^N\ln(1-s_n)\right).$$ This means the product $\prod_{n=1}^{\infty}p_n$ converges to a non-zero number if and only if $\sum_{n=1}^{\infty}\ln(1-s_n)$ converges ($\exp$ is always non-zero). In particular, we have $\ln(1-s_n)\to 0$ or equivalently $s_n\to 0$ as $n\to\infty$. Then use the fact that $\ln(1-s_n)\sim s_n$ as $s_n\to 0$.