Suppose the $f$ has an essential singularity at $z=a$.Prove that if $c\in \mathbb{C}$,and $\varepsilon >0$ are given,then for each $\delta >0$ there is a number $b$,$|c-b|<\varepsilon$,such the $f(z)=b$ has infinitely many solution in $B(a;\delta)$. This is an exercise from 'functions of one complex variable'.
I solved this question by using open mapping theorem and Baire Category Theorem to argue that $\bigcap_{i=1}^{\infty}f(\{z:0<|z-a|<1/n\})$ is a dense set.
Is there a solution which does not use Baire Category Theorem? Thank you.