Suppose $X$ and $Y$ are random variables, $E(Y^2) < \infty$ and $\varepsilon = Y - E(Y|X)$ so that $Y = E(Y|X) + \varepsilon$.
Given that $E(\varepsilon | X) = E(\varepsilon) = 0$, show that $Cov(\varepsilon , E(Y|X)) = 0$.
This question has multiple parts so $E(Y^2) < \infty$ may or may not be applicable in this case.
Here's what I tried so far. I used the fact that $Cov(X,Y) = E(XY) - E(X)E(Y)$ and $Cov(X,Y) = E[(X - E(X))(Y-E(Y))]$ and concluded that $Cov(\varepsilon , E(Y|X)) = E(\varepsilon E(Y|X))$ or in other words, $E(\varepsilon E(Y)) = 0$.
From there, I guess the only thing I have to show is that: $E(\varepsilon E(Y|X)) = 0$, but I'm having trouble doing this.
Am I going in the right track or is this completely the wrong approach to tackling this problem?