Considering a sorted sequence $a_0\leq a_1\leq...\leq a_{n-1}$ and defining the average gap of any subsequence to be:-
$f(i,j)=\frac{(aj-ai)}{(j-i)}$,
I would like to show that for all $ i=1,...,n-2$ it holds that either $f(0,i)\geq f(0,n-1)$ or $f(i,n-1)\geq f(0,n-1)$.
I am looking to prove this by contradiction, so I have assumed that $f(0,i)$ and $f(i,n-1)$ are both < $f(0,n-1)$.
Where should I proceed? Does anybody know how to prove this?