Let $x_1,x_2,x_3,...,x_n$ be positive numbers. Prove that
$$\frac{1}{1+x_1}+\frac{1}{1+x_1+x_2}+\cdots+\frac{1}{1+x_1+x_2+\cdots+x_n} < \left(\displaystyle\frac{1}{x_1}+\frac{1}{x_2}+\frac{1}{x_3}+\cdots+\frac{1}{x_n}\right)^{1/2}$$
this is what I can think of, but I don't know how to continue.
$$n=1$$ $$\frac{1}{1+x_{1}} < \frac{1}{\sqrt[ 2]{x_{1}}}$$ $$1+x_{1}>x_{1}$$ $$a)\text{ if } x_{1}>1$$ $$x_{1}>\sqrt[ 2]{x_{1}}$$ $$x+1>\sqrt[ 2]{x_{1}}$$ $$b) \text{ if }0<x_{1}<1$$ $$x_{1}<\sqrt[ 2]{x_{1}}$$ $$0<\sqrt[ 2]{x_{1}}<1$$ $$1+x_{1}>\sqrt[ 2]{x_{1}}$$ $$\therefore\ x_ {1}\text{ positive}$$ $$\frac{1}{x_{1}+1}<\frac{1}{\sqrt[ 2]{x_1}}$$
