Prove that $U=Y - E[Y|X]$ and $X$ are uncorrelated 
Let $U = Y - E[Y|X]$. How can I prove that $U$ and $X$ are not correlated?

I've been doing a lot of things but when I calculate $\text{cov}(U,X)$ I finish with $EXY - EXEY$ and not $0$ which would be the result.
Any help, guys?
Thanks

$$E[XU] = E[X(Y - E[Y|X])] = E[XY - XE[Y|X]] = E[XY] - E[XE[Y|X]]$$
Because $X$ is a function of $X$, we can pull out $X$: $XE[Y|X] = E[XY|X]$. Then $E[XE[Y|X]] = E[E[XY|X]]$, so
$$E[XU] = E[XY] - E[E[XY|X]]$$
Then $E[E[XY|X]] = E[XY]$. You can say this follows from tower property, but you can just say this is total expectation. Finally
$$E[XU] = E[XY] - E[XY] = 0$$
