For an open set $\Omega$, a function $f$ is analytic $z_0 \in \Omega$ if there exists a power series $\sum a_n(z-z_0)^n$ centered at $z_0$ with positive radius of convergence, such that:
$$f(z) = \sum_{n=0}^{\infty}a_n(z-z_0)^n$$
for all $z$ in a neighborhood of $z_0$.
There also exists a theorem that says if $f$ is holomorphic in a connected open set $\Omega$, has a zero at $z_0 \in \Omega$, and does not vanish identically in $\Omega$, then there exists a neighborhood $U \subset \Omega$ of $z_0$, a non-vanishing holomorphic function $g$ on $u$, and a unique positive integer $n$ such that
$$f(z) = (z-z_0)^ng(z)$$
for all $z \in U$.
However, does this second property hold for all $z_0 \in \Omega$ for any holomorphic $f$? Why or why not?