$f^{(k)}(z_0)=0$ for all $k\geq0$ implies that $f=0$

Question

I'm trying understand that given an analytic function in a domain, if the function and all its derivatives vanish at a point, then, the function itself is zero.

I was told that this is a direct consequence of principle of analytic continuation, but to my understanding, the principle of analytic continuation says that the zeros of an analytic function is isolated. I'm not sure how to connect the two pieces. Can anyone explain?

Answer 1

Suppose the domain is connected. You can write $f(z)=\sum a_nz^n$ in a open neighborhood $U$ of $0$ and $a_n$ is proportional to $f^{(n)}(0)$. This implies that $f$ vanishes on $U$.