Showing $f$ is constant Suppose that $f$ is an entire function and that there is a nonempty open ball $B(w_0,r)$ such that $w\notin \mathrm{Range}(f)$ for all $w\in B(w_0,r)$. Show that $f$ is constant.  
I'm not sure how to start this. This looks like maybe something to do with Liouville's Theorem, but this is under a topic where we try not to apply other theorems, so I'm not sure if I can use it here if it's even possible.

Notice that for all $z \in \mathbb{C}$, $|f(z)-w_{0}| \geq r>0$, and in particular, and hence $\frac{1}{f(z)-w_{0}}$ is entire, and bounded by $\frac{1}{r}$ (above), and hence...
