How to prove $x_1 + x_2 +\ldots+x_n \le \left(\frac{1}{2}\right)\Rightarrow (1 - x_1)(1-x_2)\cdot \ldots \cdot(1-x_n)\ge \left(\frac{1}{2}\right)$?
Proof of an inequality by induction: $(1 + x_1)(1 + x_2)...(1 + x_n) \ge 1 + x_1 + x_2 + ... + x_n$