In the problem I'm looking at we are given a hyperbolic surface contained in $\mathbb{C} \setminus \{0,1\}$ and a sequence of points $z_1, z_2,...$ that approach some point on the boundary of this surface.
We can construct a family of covering maps $p_n : \mathbb{D} \rightarrow \mathbb{C} \setminus \{0,1 \}$ such that $p_n(0)= z_n$ and $p_n (B(0,r)) = B(z_n, r)$ for some $r>0$ where $B(,)$ is a ball.
I want to show that the spherical diameters of these balls approach zero. Here's one of the arguments made in the proof I'm looking at:
"If they didn't approach zero, one could apply Montel's Theorem onto the subsequence that doesn't approach zero and that would imply they would converge locally uniformly to a constant map, which is impossible."
However, no further justification is made here and I'm a little confused on two things:
Why must it approach a constant map?
Why is it impossible for a subsequence of covering maps to converge locally uniformly to a constant map?
Any help would be greatly appreciated. (The link to the proof I'm looking at can be found on page 23/24 here - https://legacy-www.math.harvard.edu/theses/senior/dozier/dozier.pdf)