Let $n$ be a positive integer with $n>1$, and let $a_1$, $a_2$, ..., $a_n$ be positive integers such that $a_1<a_2<...<a_n$ and $\frac{1}{a_1}+\frac{1}{a_2}+...+\frac{1}{a_n} \le 1$. Prove that $(\frac{1}{a_1^2+x^2}+\frac{1}{a_2^2+x^2}+...+\frac{1}{a_n^2+x^2})^2 \le \frac{1}{2} \cdot \frac{1}{a_1(a_1-1)+x^2}$ for all real numbers $x$.