Suppose that $f$ is a meromorphic function on the punctured disk $0<|z|<1$ having poles on the set $\left\{\frac{1}{k}: k\in\mathbb{N}\right\}$. I would like to show that for every $1>r>0$, the restriction of $f$ to $0<|z|<r$ has a range that is dense in $\mathbb{C}$.
I know that if $f$ had an essential singularity at the origin instead of this dense set of poles, then the Casorati–Weierstrass theorem would guarantee that the range of $f$ is dense in any punctured disk.
How to deal with this dense set of zero though?