This comes from the book "A Probability Path". I'm just working through the problems trying to get a grasp of conditional expectations.
Suppose $X,Y$ are random variables with finite second moments such that for some decreasing function $f$ we have $E(X|Y)=f(Y)$. Show that $\operatorname{Cov}(X,Y)\leq 0$.
I can't figure out how to relate $E[X|Y]$ to $E[XY]$ which was my approach. If I could get a hint for this or how the decreasing function affects $E[X],E[Y],E[XY]$. Is there some work with bounding involved?