I am stuck on a note by Conway right before the proof of Marty's Theorem (equivalent of Montel's Theorem, for meromorphic functions) in "Function of One Complex Variable I".
The family of functions given as example is $$f_n(z) = nz.$$
The set $\mathcal F = \{f_n\}$ is relatively compact in $C(G,\mathbb C_\infty)$, where $G$ is a region. In fact we have that $f_n \to f \equiv \infty$. So by the Theorem the family $\mu(\mathcal F) = \{\mu(f) : f \in \mathcal F\}$ is locally bounded.
We have that the spherical derivative of $f_n$ is $$\mu(f_n)(z) = \frac{2n}{1+n^2|z|^2}.$$
According to what I understood local boundedness means that for any point $a\in G$ there exists $r$ and $M$ such that $$|\mu(f_n)(z)| \leq M$$ for all $z$ such that $|z-a|<r$, for all $f_n\in \mathcal F$.
However I don't see how to find such an $M$ if I take $a=0$, since for $z=0$ $$\mu(f_n)(0) = 2n.$$
Thanks for your help in clarifying this point.
$f_n(z) = nz$ converges to $f \equiv \infty$ only for $ z \ne 0$, and $\mathcal F = \{f_n\}$ is relatively compact in $C(G,\mathbb C_\infty)$ only for regions $G \subset \Bbb C\setminus \{ 0 \} $.
Any compact subset $K$ of such a region $G$ has a positive distance $r$ from origin, so that $$ \mu(f_n)(z) = \frac{2n}{1+n^2|z|^2} \le \frac{2n}{1+n^2 r^2} \le \frac 1r $$ for all $z \in K$ and all $n$, i.e. the spherical derivative is indeed locally uniformly bounded.