I am having the hardest time proving this inequality:
Let $n$ be a natural number, and $a_1,a_2,\cdots,a_n$ and $b_1, b_2, ..., b_n$ be nonnegative numbers such that $\sum_{i=1}^n a_i = \sum_{i=1}^n b_i = 1$. Then $$\sum_{i=1}^n \sqrt{a_i b_i} \le 1 - \frac{\big(\sum_{i=1}^n |a_i - b_i|\big)^2}{8}.$$
The proof must be related to Cauchy-Schwarz inequality, but I seem not to be able to connect the dots. Any idea?