Suppose that $\displaystyle f(z)$ is analytic at $\displaystyle z_0$.

Prove that a necessary and sufficient condition that $\displaystyle z_0$ is a zero of order $\displaystyle m$ is that $\displaystyle f(z)$ can be written as

$\displaystyle \begin{align*}f(z)=(z-z_0)^m g(z)\end{align*} ,$

where $\displaystyle g(z)$ is analytic and nonzero at $\displaystyle z_0$.

The proof seems intuition, but I don't know where to start. Thank you.