}
Let M be 
with the unit circle shrunk to
a point P.
Again M is topologically a sphere,
so we may give it a bounded metric d.
Let 
be the set of sequences 
which converge to P and define a metric 
on 
by
Let the group 
of permutations of 
act on
by permuting the coordinates.
Define a metric D on the quotient space 
by
letting
Here 
denote the projections of
to 
.
(That 
if and only
if 
requires the convergence of
x,y.)
The set of zeros of a power series
inside 
forms a sequence in M converging to
P
(by Proposition 2.1) or else is finite, in which case we
append an infinite sequence of P's.
This defines a map
By the same Rouché's theorem argument used in the proof of
Theorem 4.1, this map is continuous.
The conditions of Lemma 4.1 hold for the same reason as
before, so the image of f is path connected.
Suppose 
.
Let 
be a path from the image
under f of a 0,1 power series vanishing at 
to
.
Fix 
, and let 
be 
with the
annulus 
shrunk to a point Q.
Define 
on 
by letting 
.
By Proposition 2.1 there is an upper bound n on the
number of zeros of a 0,1 power series inside 
.
The path 
induces a path
(Apply the projection 
to each element of 
,
and throw away infinitely many Q's to get 
.)
Pick 
such that 
.
We define inductively a sequence of paths
each extending the one before.
First apply Lemma 5.1 to lift 
to a path
.
Since some coordinate of 
is 
and since
all
coordinates of 
are Q, we get a path
from 
to Q in 
.
Let 
be the smallest 
such that
.
Then by restriction to 
we get a path
in 
since
can be identified with
.
Finally, since 
is always a coordinate of
, 
for all 
.
By the same process, we inductively find for each 
a path
such that
, let
be the smallest 
such that
and obtain a path
which we append to 
to obtain
such that 
is always a coordinate of 
.
Let 
.
Piecing together the 
's gives a continuous map
such that 
is a coordinate of 
for all
.
The set of limit points of 
as 
is a closed interval I.
Let 
.
If 
is distinct from 
then 
(since
) and by continuity of 
, r also
differs by some 
from all coordinates of 
for t
in a neighborhood of 
, so r cannot be a limit point of
as 
.
Thus 
but 
so I must be a single point.
Since 
for 
,
.
Case 1:
1 is the only limit point of 
as 
.
Then 
extends to a path 
from 
to 1.
Case 2:
There is a limit point 
, 
, of
as 
.
By Theorem 3.1, there is an open disc centered at
contained in 
.
For some 
, 
is in this disc, so we
can replace the tail end of 
on 
by a
straight line from 
to 
in
.
In either case we can connect 
to a point on the unit
circle via a path in 
.
The same is true if 
,
, since 
is closed under 
.
Since 
contains the unit circle, this proves that
is path connected. 
