Hi. I'm trying to solve the second statement.

My try:

$|T(F)-T(G)| = |\int x dF - \int x dG| = |\int x d(F-G)|$ $\leq \int |x|d(|F-G|) \leq M \int d(|F-G|) \leq M \sup_{x}|F(x)-G(x)|$

However, I can not verify the last inequality, rather is it true?

This is Exercise 2.5 of "All of Nonparametric Statistics, Larry Wasserman"

I'm trying to solve the second statement of the following exercise. It is Exercise 2.5 of "All of Nonparametric Statistics, Larry Wasserman".

My try:

\begin{align} |T(F)-T(G)| &= \left|\int x dF - \int x dG\right| \\ &= \left|\int x d(F-G)\right| \\ &\leq \int |x|d(|F-G|) \\ &\leq M \int d(|F-G|) \\ &\leq M \sup_{x}|F(x)-G(x)| \end{align}

I can not verify the last inequality. Is it true?