$\prod\left(1-p_n\right)>0$ 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?

Why it is sufficient to prove the hint: Suppose $\sum p_n < \infty$. Then there is an integer $N$ such that $\sum_{n \geq N} p_n < 1$. Now observe that both $\prod_{n < N} (1-p_n)$ (a finite product) and $\prod_{n \geq N} (1-p_n)$ (using the hint) are positive.
