Let $u:\mathbb R^2 \to \mathbb R$ be a non-surjective harmonic function.
$(i)$ Show that $u$ is bounded from below or from above.
$(ii)$ Prove that $u$ is constant (and therefore any harmonic function is constant or surjective)
I know how to prove $(i) \implies (ii)$:
Suppose $|u|\leq B$, by hypothesis, there exists an entire function $f$ such that $f(x+iy)=u(x,y)+iv(x,y)$. If I consider $e^{f(z)}$, then $e^{f(z)}$ is an entire function and $$|e^{f(z)}|=|e^{u+iv}|$$ $$=e^u|e^{iv}|$$ $$\leq e^B.1$$ $$=c \in \mathbb R_{\geq 0}$$
This proves $e^{f(z)}$ is bounded so, by Liouville's theorem, $e^{f(z)}$ is constant. In this exercise Application of Liouville's theorem exercise it is proven that $e^{f(z)}$ constant $\implies$ $f(z)$ constant.
I am having problems with the first part ($(i)$) of the exercise. Suppose $u$ is not surjective, then there exists $u_0 \in \mathbb R$ such that $u(x,y) \neq u_0$ for all $(x,y) \in \mathbb R^2$. I have no idea how to deduce $u$ is not bounded from this hypothesis, I would appreciate any hints.