For
, we have
We already know the limit (call it ) exists, that
,
and that
by Lemma II.45 and Proposition II.44. Suppose, by way of contradiction, that
.
This implies that
.
Let
.
Define
by
.
(Note:
.
For
,
no problem. Otherwise,
implies
implies
implies
.)
We remark that is uniformly continuous on because the inversion map of
to itself is continuous and is compact. Given
and th roots
,
,
…,
of unity, we have
.
(This is a polynomial identity.)
Replace by to get
.
Multiply by to get
.
By Lemma II.46,
.
All the factors commute, so for
,
we can multiply both sides by
to obtain
Given
there exists
such that
whenever
.
Take with
and
.
We have
,
and hence
,
for
.
It follows from Equation II.B that
Equation II.C holds for every . Because
we have
for sufficiently large . Thus
,
.
So
and
.
By Equation II.C,
for sufficiently large , so
.
Notice
.
But
so
implies
.
However,
for all ,
i.e.
for all , which is a contradiction.