Criterion of flatness for projective morphism

Question

Let $f: X \to Y$ be a projective morphism, $\mathcal O(1)$ is a relatively very ample sheaf, then f is flat iff $f_*\mathcal O(m)$ is locally free for big $m$. I can prove flatness implies that $f_*\mathcal O(m)$ is locally free for big $m$, but I have no idea how to prove the other side of the statement.