$\prod\left(1-p_n\right)>0$ I want to prove that if $0\le p_n<1$ and $\sum p_n<\infty$, then $\prod\left(1-p_n\right)>0$ .
There is a hint : first consider the case $\sum p_n<1$, and then show that $\prod\left(1-p_n\right)\ge1-\sum p_n$ .
How can I use this hint to show the statement above?

Suppose $\sum p_n = C$. Let's notice that there exists $N: \sum_{k=1}^{N}p_n > C-1$. Let's replace N first numbers with $0$. $\prod(1-p_n) = c\prod_{n > N}(1-p_n)$ We know from hint $\prod_{n > N}(1-p_n) > 1-\sum_{n > N} p_n > 0$. Then $\prod(1-p_n) = c\prod_{n > N}(1-p_n) > 0$.
