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

Since $1-x\geq \exp\left(-\frac{x}{1-x}\right)$ for any $x\in[0,1)$, it follows that:
$$\prod(1-p_n)\geq \exp\left(-\sum \frac{p_n}{1-p_n}\right),$$
but since $\sum p_n$ is converging, it follows that for any $n$ big enough ($n\geq N$) we have $p_n\leq\frac{1}{2}$, so:
$$\prod(1-p_n) \geq \exp\left(-\sum_{n=1}^{N}\frac{p_n}{1-p_n}-2\sum_{n>N}p_n\right)>0.$$
