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
. 
