Prove $Cov(X,Y)=0$ if $E(X\mid Y=y)$ is a constant

Question

Let $X$ and $Y$ be two random variables. Suppose $E(X\mid Y=y)$ is a finite constant $c$. Show that $Cov(X,Y)=0$.

What I know:

$Cov(X,Y)=E(XY)-E(X)E(Y)$.

$$E(X|Y=y) = \sum_{x} xp_{X|Y}(x/y)$$

I am give that the lower term is a constant but nothing more.

I know that $$E(X|Y=y) = E(X)$$ when both X and Y are independent but I think that is useless here becasue we are not given that $ X$ and $Y$ are independent.

Answer 1

$\mathbb{E}[X]=\mathbb{E}[\mathbb{E}[X|Y]]=C$

$\mathbb{E}[XY]=\mathbb{E}[\mathbb{E}[XY|Y]]=\mathbb{E}[Y\mathbb{E}[X|Y]]=C\mathbb{E}[Y]=\mathbb{E}[X]\mathbb{E}[Y]$