.
Since is closed under and , it suffices to show that for |z| <1, , and |z+1| sufficiently small, z is in . (Proposition 3.1 handles the case .) Let . Let B be the parallelogram with vertices .We work in a nonstandard coordinate system for , with basis vectors 1 and , so B is represented by the square with vertices . We claim that multiplication by z is represented by the matrix up to . We have
and so corresponds to in our basis, and is .>From Lemma 3.5, it follows then that
so for sufficiently small , we may apply Lemma 3.1 to deduce .