Suppose that X and Y are random variables on a common probability space such that $E(X^2+Y^2) \lt \infty$, $E(X|Y)=Y$, $E(Y|X)=X$. Prove that $$P(X=Y)=1$$
My work: $E(X)=E(E(X|Y))=E(Y)$ and $E(Y)=E(E(Y|X))=E(X)$
But I don't know what to do next and I'm sure how to use the condition $E(X^2+Y^2) \lt \infty$.
Thanks in advance.