Chapter 8, Problem 8.15STPE

Prove Chebyshev’s sum inequality, which says that if $a_1\ge a_2\ge \mathrm{...}\ge a_n$ and $b_1\ge b_2\ge \mathrm{...}\ge b_n$, then $n{\displaystyle {\sum }_{i=1}^n{a}_i{b}_i}\ge \left({\displaystyle {\sum }_{i=1}^n{a}_i}\right)\left({\displaystyle {\sum }_{i=1}^n{b}_i}\right)$.