Originally published at 狗和留美者不得入内. You can comment here or there.
Let’s first state it.
Theorem (Hurwitz’s theorem). Suppose is a sequence of analytic functions on a domain
that converges normally on
to
, and suppose that
has a zero of order
at
. Then for every small enough
, there is
large such that
has exactly
zeros in the disk
, counting multiplicity, and these zeros converge to
as
.
As a refresher, normal convergence on is convergence uniformly on every closed disk contained by it. We know that the argument principle comes in handy for counting zeros within a domain. That means
The number of zeros in ,
arbitrarily small, goes to the number of zeros inside the same circle of
, provided that
.
To show that boils down to a few technicalities. First of all, let be sufficiently small that the closed disk
is contained in
, with
inside it everywhere except for
. Since
converges to
uniformly inside that closed disk,
is not zero on its boundary, the domain integrated over, for sufficiently large
. Further, since
uniformly, so does
, so we have condition such that convergence is preserved on application of integral to the elements of the sequence and to its convergent value. With
arbitrarily small, the zeros of
must accumulate at
.