How to show $x_1,x_2, \dots ,x_n \geq 0 $ and $ x_1 + x_2 + \dots + x_n \leq \frac{1}{2} \implies (1-x_1)(1-x_2) \cdots (1-x_n) \geq \frac{1}{2}$

It is easy to see that:

$$(1-a)(1-b) \geq 1-(a+b)$$

Then, you can use induction to prove that:

$$(1-x_1)(1-x_2)...(1-x_n) \geq 1-(x_1+x_2+...+x_n)$$

The inductive step is:

$$(1-x_1)(1-x_2)...(1-x_n)(1-x_{n+1}) \geq \left[ 1-(x_1+x_2+...+x_n) \right] (1-x_{n+1}) \geq 1-(x_1+x_2+...+x_n+x_{n+1})$$

For this to work you only need that all $1-x_i \geq 0$...Of course you need $x_1+..+x_n \leq \frac{1}{2}$ to get the desired inequality.