We are given that $x_1, x_2, \cdots, x_n$ are positive real numbers and $x_1x_2\cdots x_n\geq 1\forall n\in \mathbb N$.
We have to show $(1+x_1)(1+x_2)\cdots (1+x_n)\geq 2^n$.
For $n=1$ the result is trivial as $x_1\geq 1\Rightarrow (1+x_1)\geq 1+1=2^1.$
Let the result be true for $n=m$ viz $x_1x_2\cdots x_m\geq 1$ implies
$$(1+x_1)(1+x_2)\cdots (1+x_m)\geq 2^m$$
Multiplying both sides by the inequality $1+x_{m+1}\geq 2$ we see that
$$(1+x_1)\cdots(1+x_m)(1+x_{m+1})\geq 2^{m+1}$$ which shows that the inequality is true for $n=m+1$ whenever it is true for $n=m$.
Induction now completes.