Inequality in Algebra: $1 \leq x_1 x_2 \cdots x_n$ implies that $2^{n} \leq (1 + x_1)(1+x_2) \cdots (1 + x_n).$ How do I prove that if $x_1, \ldots, x_n$ are positive real numbers, then
$$1 \leq x_1 x_2 \cdots x_n \text{ implies that } 2^{n} \leq (1 + x_1)(1+x_2) \cdots (1 + x_n).$$
I attempted a proof by induction but am not able to nail the inductive step. Any help would be appreciated!

Hint 
$$x_{i}+1\ge 2\sqrt{x_{i}}$$
