Image of $f$ comes arbitrarily close to every $c$ in $\mathbb{C}$.

Question

Let $p_1, p_2,\ldots,p_n$ are points on the compact Riemann surface $X$ and $Y:=X-\{p_1,.....,p_n\}$. Suppose $$f:Y→\mathbb{C}$$ is a non-constant holomorphic function. Show that the image of $f$ comes arbitrarily close to every $c$ in $\mathbb{C}$.$//$ Is this mean that, we have to show $f(Y)$ is dense in $\mathbb{C}$? Then how can we proceed? Please help me.

Answer 1

Let's say the image misses some ball $B(z,r)$. Then we consider the function $g(z) = \frac{1}{f(z) - c}$. Then $g$ is non-constant and bounded by $\frac{1}{r}$, so we may extend it to obtain a non-constant holomorphic function on $X$, which is impossible.