Holomorphic function convergent sequence

[binbagsss]

Homework Statement

[/B]
Hi

Theorem attached and proof.

I am stuck on

1) Where we get $|g(z)|\geq |a_m|/2$ comes from
so $a_{m}$ is the first non-zero Fourier coeffient. So I think this term is $< |a_m|r^{m}$, from $r$ the radius of the open set, but I don't know how to take care of the rest of the higher tems through $a_{m}$ , is this some theorem or?

2) The conclusion thus $f(z)$ has only one zero at $z=z_0$
I think I'm being stupid but what is this being made from?
We know $g(z_0) = a_{m} \neq 0$ and $a_{0}=0$, but I don't understand.

Thanks

Homework Equations

above

The Attempt at a Solution

above

 

[andrewkirk]

The Lemma is very badly expressed. What do you think it means by 'if $f(z_n)=0$ for any $n$, then...'? This vague phrase could either mean
(1) ''if there exists some $n\in\mathbb N$ such that $f(z_n)=0$, then..."
or it could mean
(2) "if for every natural number $n$, $f(z_n)=0$, then ..."

The two interpretations have very different consequences.