I am trying to understand the proof of Picard's big theorem which is theorem $4.20$ in Wilhelm Schlag's book A course in complex analysis and Riemann surfaces.
The theorem is stated as follows
If $f$ has an isolated essential singularity at $z_0$, then in any small neighborhood of $z_0$ the funtion $f$ attains every complex value infinitely many ofther with one possible exception.
Schlag claims this is an application of Montel's normality test which says
Any family of functions $\mathcal{F}$ in $\mathcal{H}(\Omega)$ which omits two distinct values in $\mathbb{C}$ is a normal family.
Schlag proves the big Picard theorem in a few lines as follows:
Let $z_0=0$ and define $f_n(z) = f(2^{-n}z)$ for an integer $n\geq 1$. We take $n$ so large that $f_n$ is analytic on $0<\lvert z\rvert < 2$. Then $f_{n_k}(z)\rightarrow F(z)$ uniformly on $1/2\leq \lvert z\rvert \leq 1$ where either $F$ is analytic of $F\equiv\infty$. In the former case, we infer from the maximum principle that $f$ is bounded near $z=0$, which is therefore removable. In the latter case, $z=0$ is a pole.
I understand the steps that are taken in the proof but I am confused as to how it proves the statement. I guess he uses some sort of contraposition and assumes that $f$ omits two values in a neighborhood of $z_0$. But I am not sure how that shows $f$ hits every complex value infinitely often with one exception (which can be reached finitely many times).
He is proving that a function holomorphic in a annulus $\{z:0<|z|<R\}$ which omits two values has either a removable singularity or a pole at $z=0$, and certainly does not have an essential singualarity there.