Suppose that $u_n$ is a sequence of harmonic function on an open, connected subset $D \subset \mathbb{C}$ such that $u_n(z) \in (0, \infty)$ for all $z \in D$ and with $u_n(z_0) \to 0$ for some $z_0 \in D$. Show that $u_n \to 0$ uniformly on compact subset of $D$.
Thoughts so far: I'm new to harmonic functions, so I'm not sure where to begin. I know that locally each $u_n$ is the real part of a holomorphic function.
Context: I'm studying for a qual, so just a hint to get me going or any ideas to get started would be most helpful.
Consider first the case of a disk $B$ containing $z_0$. Then $u_n = \text{Re}(f_n)$ where $f_n$ is analytic on $B$. By Montel's theorem the $f_n$ form a normal family. What can you say about the limit of a subsequence that converges on $B$?