I am stuck on a question about conditional expectation, I was wondering if someone could give me a hint so I can make a start.
Let $X$ and $Y$ be random variables with finite mean and let $\mathbb{E}[X^2]<+ \infty$. Proof that $\mathbb{E}[X^2|Y]\geq (\mathbb{E}[X|Y])^2$ almost surely.
I have tried to write out something with the conditional variance, but this quickly showed that it did not work out. I don't really know how to continue. Any hints would be greatly appreciated, thanks in advance!