Let $f$ be analytic in $G$ ={$z:0<|z-a|<r$} except that there is a sequence of poles {$a_n$} in $G$ with $a_n\to a$. Show that for any $w$ in $\mathbb{C}$ there is a sequence {$z_n$} in $G$ with $a=$ lim $z_n$ and $w=$ lim $f(z_n)$.
I was trying to solve some problems of Conway complex analysis but I can not able to solve this problem. I was trying to look at the behavior of $f$ around $a$ and if I able to show that $a$ is essential singularity then we are done. But $a$ is not an isolated singularity also. Any help/hint in this regards would be highly appreciated. Thanks in advance!