Prove that if $0\le p_n< 1$ and $S=\sum p_n< \infty$ then $\prod (1-p_n)>0$. Hint given: First show that if $S<1$, then $\prod (1-p_n)\ge 1-S$.
Attempt: I was able to show the hint by using recursion setting $A_n=\prod_{i=1}^{n}(1-p_i)$ re expression the $\prod (1-p_n)$ as $1-S+\sum_{n_1, n_2=1,n_1