Proposition 2.2

  There exists a sequence of points such that as and

 

Proof

Consider the polynomial

 

with . For n large compared to m, we will show that has a zero near to

 

and . We show that one can take .

To show that has a zero near , let

 

Then . Consider the circle . On this circle, , while

 

so for , by Rouché's theorem and have the same number of zeros inside the circle, namely one. This proves the claim and answers the Conway-Parker question.