I have proved that
if the radius of convergence for $f(z)=\sum_{n=0}^{\infty}a_nz^n$ is positive and not all $a_n$ are Zero then there is a circle around $0$ which has at most finitely many zero points in $f$.
How can I derrive from that the identity Theorem of Powerseries which is:
The powerseries
$$f(z)=a_0+a_1z+a_2z^2+...$$
$$g(z)=b_0+b_1z+b_2z^2+...$$
both have radius of convergence $\neq 0$.
If $(z_k)$ is a Zero convergent series with $z_k\neq 0$ and $(z_k) = g(z_k)$ for all $k$.
I don't understand what's written in parethesis and what follows now
(For example $f(z)=g(z)$ in a disk around $0$)
Then $a_n=b_n$ for all $n=1,2,...$
I don't understand how I can derive from the previous result this result like I have already said, in the book there is also no further explaination. They have described the Identitys-theorem just as an application of the first Theorem I have stated.
Please help me