Let $f$ be a holomorphic function on an open, connected set $\Omega\subset \mathbb{C}$ with $z_0\in \Omega$ a fixed point, and $\{f^n\}_{n\in \mathbb{N}}$ the sequence of iterates.
I want to prove that if |f'(z_0)|<1 then there is a neighborhood of $z_0$ such that $\{f^n\}$ is normal in it.
I don't really know what to do, because I don't know how to handle the iterates. I think the fact that $|f(z)| < |z-z_0| + |z_0| + \epsilon |z-z_0|$ might be useful to apply Montel's theorem, but I didn't get very far.
This is not homework, it's a problem I'm trying to solve as preparation for a complex analysis exam.
Also: I'd be grateful to have a geometrical interpretation of |f'(z_0)|<1 or of |f'| in general. I don't really understand what does it mean for |f'(z_0)|<1, for example in Schwarz's lemma.