convex polytopes – Lattice points of a subset of [0,1]^n

Given a set of ${0, 1}^{n}$ vectors (say $S$), I am interested in the lattice points of $conv(S)$, their convex hull. It seems pretty obvious to me that the the set of $conv(S)$‘s lattice points is just $S$ itself, but I don’t know how to really prove it.