Let $F(x)$ be a CDF. Let $U_i$ ($1\le i \le n$) be i.i.d. random variates chosen from the standard uniform distribution on $(0,1)$. Let $U_{(i)}$ be the $i^{th}$ order statistic, i.e., $i$ of the $U$s are less than or equal to $U_{(i)}$. Let $F_n(x)$ be empirical CDF of the $U$s, i.e., $F_n(U_{(i)})=i/n$. Now, consider: $$ E\lvert (F_n(U_{(i)})-U_{(i)})F^{-1}(U_{(i)})\rvert $$ where $E$ is the expectation operator.
I'm reading a paper which applies Hölder's inequality to this term, using $p=\infty$ and $q=1$ and obtains: $$ E\lvert (F_n(U_{(i)})-U_{(i)})F^{-1}(U_{(i)})\rvert \le E \operatorname{ess}\sup_{U(i)} \lvert F_n(U{(i)})-U_{(i)} \rvert E \lvert F^{-1}(U_{(i)}) \rvert $$ I believe I understand Hölder's inequality, and essential suprema, but I don't understand the conclusion. What does the expectation of that supremum mean? Is the expectation operator spurious?