Corollary 3.1
If 
for sufficiently small 
then z is a multiple zero of some 0,1 power series.
Proof
By Theorem 3.1,
if 
is small enough we can pick 0,1 power series 
and zeros 
of 
such that
and 
as 
.
By taking a subsequence we may assume that the coefficient of 
in 
is eventually constant for large n,
for each k.
By a Rouché's Theorem argument, the pairs of zeros
of 
must converge to (at least)
a double zero at z of 
. 