Let $x_1,x_2,\ldots, x_n\in \Bbb R^+\cup \{0\}$ such that $x_1+\cdots +x_n\le 1/2$. Prove that $\displaystyle \prod_{i=1}^n(1-x_i)\ge 1/2$.
It is easy to prove it by induction but I have a doubt since with Bernoulli's inequality it is also done by induction but it can be done with AM-GM but in this one I have not been able to do it, whether it will be possible or not? Some help?