A function is isotonic if for all .
A sequence of points is called -grid if , , and for all , we have . Note this imply .
Let be a -grid of points where . is a isotonic function such that and , then
It’s easy to see that is a increasing function. Let the points in ordered as . Let . Note that
thus by isotonic function . This is just .
for all , because , thus .
Combine the relations above, we have
But , so , . Since is small, we have .