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.