Let $a_1 \leq a_2 \leq \cdots \leq a_n$ and $b_1 \leq b_2 \leq \cdots \leq b_n$, then prove that $$n\sum_{i=1}^n a_ib_i \geq \sum_{i=1}^n a_i \cdot \sum_{i = 1}^n b_i.$$
Attempt
The $n\displaystyle \sum_{i=1}^n a_ib_i$ makes me think of Cauchy-Schwarz but I am not sure how to use it. I could use AM-GM to say $n\displaystyle \sum_{i=1}^n a_ib_i \leq n\displaystyle \sum_{i=1}^n \dfrac{(a_i+b_i)^2}{4}$, but I am not sure how to relate that to the other terms.