Let $a_k, b_k (k=1,2,...,n)$ are real positive numbers. Prove the inequality $$\frac{a_1b_1}{a_1+b_1}+\frac{a_2b_2}{a_2+b_2}+...+\frac{a_n b_n}{a_n+b_n}\le \frac{AB} {A+B}$$ where $A=a_1+a_2+...+a_n$ and $B=b_1+b_2+...+b_n$
My work so far:
If $n=2$ $$\frac{a_1b_1}{a_1+b_1}+\frac{a_2b_2}{a_2+b_2}\le \frac{(a_1+a_2)(b_1+b_2)}{a_1+a_2+b_1+b_2} \Leftrightarrow$$ $$\Leftrightarrow (a_1b_2-a_2b_1)^2\ge0$$