a set is infinite iff there is a bijection between it and a proper subset

I've been trying to prove that a set S is infinite if any only if there is a proper subset S' and a bijection $\displaystyle \phi : S \longleftrightarrow S'$.

Assuming the bijection exists, it's pretty clear that S cannot be finite, and it was easy to prove this. But I'm not having much luck going the other direction:

Assuming S is not finite, prove there exists $\displaystyle S' \subset S$ with a bijection $\displaystyle \phi : S \longleftrightarrow S'$.

Any tips?