China Mathematical Olympiad

For a given positive integer $n\ge 2$, suppose positive integers $a_i$ ($i=1,2,\dots ,n$) satisfy $a_1<a_2<\cdots <a_n$ and ${\sum }_{i=1}^n\frac{1}{a_i}\le 1$. Prove that, for any real number $x$, the following inequality holds, $${\left[\sum _{i=1}^n\frac{1}{a_i^2+x^2}\right]}^2\le \frac{1}{2}\cdot \frac{1}{a_1(a_1-1)+x^2}.$$