Suppose that there exists a constant $C$ such that the following relation holds for all $G$: \begin{equation*} \vert T(F)-T(G) \vert \le C \sup_y \vert F(y)-G(y) \vert \end{equation*} Suppose that $|Y|\le M < \infty$. Show that $T(F)=\int ydF(y)$ satisfies the previous relation.
In particular we are in the setting of nonparametric statistics. So T(F) is a statistical functional. F(y) and G(y) are the real cumulative distribution function that we would like to estimate.
It is an exercise in the second chapter of "all of nonparametric statistics" by Wassermann.
I have difficulties setting up this problem and understanding how to solve it. Can somebody help please? Thanks in advance!