$E[X\mid Y]=E[X]=0$ implies $\operatorname{Cov}(X,Y)=0$

Question

So how can I prove that? I know that $\operatorname{Cov}(X,Y)=E[XY]-E[X]E[Y]=E[XY]+0$ but I am not sure what to do next? I am completely new to this so please be as basic as possible.

Thank you.

Answer 1

HINT

In general, a good identity to remember is that $$ E(Z) = E[E(Z\mid Y)].$$ So we can apply this to $Z=XY$ to give $$ E(XY) = E[E(XY\mid Y)].$$ Now what can we do with $E(XY\mid Y)$ to finish the problem?