Let be $x_1, x_2, \ldots , x_n$ strictly positive real numbers. Prove that the following inequality holds:
$$\frac1{1+x_1}+\frac1{1+x_1+x_2}+\cdots+\frac1{1+x_1+x_2+\cdots+x_n} < \sqrt{\frac1{x_{1}}+\frac1{x_2}+\cdots+\frac1{x_n}}$$
How may I tackle this inequality? I tried AM-GM, but it seems of no help.