In Ullrich's Complex Made Simple, Proposition $3.14$ part $ii$ states
If $f \in H(D'(z,r))$ (holomorphic in the punctured disk) has an essential singularity at $z$ if and only if $f(D'(z,\rho))$ is dense in $\mathbb C$ for all $\rho \in (0,r)$.
I understand the if part which refers as Casorati–Weierstrass theorem. But in the proof to only if part I am having trouble to fill a gap. The proof proceeds by contradiction.
If $f$ has a pole or removable singularity at $z$ then $f(w)$ has (finite or infinite) limit as $ w \to z$. [This part is OK. The limit would be $0$ if $z$ is removable singularity and infinite if $z$ is a pole.] Then it claims the range $f(D'(z, \rho))$ would be obviously not dense. I am having trouble to understand the 'obvious part'.