Proof of a Cantor-Bernstein-like theorem
If $A, B$ are sets and there exists an injective function $f : A \to B$ and a surjective function $g: A \to B$, does this imply there is a bijective function $h : A \to B$?
If not is there a simple counter example?
I was wondering because while reading about Cantor's Theorem, it seems like an injective function existing is like saying $|A| \le |B|$, and surjective function existing says $|A| \ge |B|$. So when they both exist I would expect a bijection to exist too.
Am I correct in my reasoning?