First we show that for 
,
is connected.
The idea is to apply Lemma 4.1 to the function f which assigns to
the set of zeros of
inside 
.
To make a continuous map of this requires some manipulation.
By Jensen's theorem, as was shown in Section 2, there is an upper bound n on the number of zeros that a power series
with 0,1 coefficients can have
inside 
.
Let M be 
with the annulus
shrunk to a point P.
(Therefore M is topologically a sphere.)
To each power series 
,
,
we assign the set of zeros inside 
,
(counted with multiplicities) and throw in extra
copies of the point P as necessary to bring the total number of points
to n.
Since the order of these n elements of M is unspecified, we obtain a point
of 
.
Let 
be this point.
We claim that this map
is continuous.
This follows easily from Rouché's theorem;
if two power series agree in the first m coordinates for m
sufficiently large then their zeros inside
will be within 
.
Some may escape or enter the disk, but this is not a problem, since in the topology on M,
P is close to all points
z with |z| sufficiently near 
.
We next check condition 
of Lemma 4.1.
This is easily done using the following trick:
given 
, let
.
Then 
,
, and 
(we extend v,w to infinite
vectors by appending 0's), since
and
have the same zeros inside 
.
Therefore we may apply Lemma 4.1 and deduce that the image
of f is path connected.
Since 
,
we may apply Lemma 4.2 with 
to deduce that
with the annulus
shrunk to a point P is a connected
subset of M.
This is equivalent to the connectivity of 
.
Since 
is the decreasing
intersection of the compact connected sets 
,
it too is connected.
So is its image under 
.
Finally, 
is the union of these two sets,
which meet on the unit circle, so 
is connected as well. 
