SMO 2014 Problem 3

Let $0<a_1<a_2<\cdots <a_n$ be real numbers. Prove that $$\left (\frac{1}{1+a_1}+\frac{1}{1+a_2}+\cdots +\frac{1}{1+a_n}\right )^2 \leq \frac{1}{a_1}+\frac{1}{a_2-a_1}+\cdots +\frac{1}{a_n-a_{n-1}}.$$