Prove inequality $(1+\frac{1}{2n})^n<2$

You can prove by induction that $$(1-x_1)(1-x_2)\cdots (1-x_n)\geq 1-x_1-x_2-\ldots -x_n$$ for all real numbers $x_1,x_2,\ldots,x_n\in[0,1]$ (the equality holds, by the way, iff at least $n-1$ of $x_1,x_2,\ldots,x_n$ are $0$). Using that inequality, we have $$\frac{1}{\left(1+\frac1{2n}\right)^n}=\left(1-\frac{1}{2n+1}\right)^n\geq 1-\frac{n}{2n+1}>\frac{1}{2}.$$