Is the family $$\mathcal F=\left\{\frac {1}{z-q}:q \in \mathbb Q\right\}$$ of meromorphic functions normal?
Relevant definitions:
1. Let $\{f_n\}$ be a sequence of meromorphic functions on a domain $D$ and $z_0 \in D.$ Then $z_0$ is said to be a $C_0-$ point of $\{f_n\}$ if there exists a disk $D_0 \subseteq D$ such that $\{f_n\}$ is uniformly convergent in $D_0$ with respect to spherical distance.
2. A sequence $\{f_n\}$ is said to be $C_0$ in a domain $D$ if each point of $D$ is a $C_0-$ point of $\{f_n\}.$
3. A family $\mathcal F$ is of meromorphic functions in a domain $D$ is said to be normal in $D$ if for every sequence in $\mathcal F,$ there exists a subsequence which is $C_0$ in $D.$
Relevant theorems:
1. A family of meromorphic functions in a domain $D$ is normal in $D$ if and only if it is equicontinuous in $D$ with respect to spherical distance.
2. Let $\mathcal F$ be a family of meromorphic functions in a domain $D.$ Then $\mathcal F$ is normal in $D$ if and only if for each $z \in D$, there exists a disk $D_0$ with $z \in D_0 \subseteq D$ and $M>0$ such that for each $f \in \mathcal F$ $$\text{either }|f|<M \text{ or } \frac{1}{|f|}<M$$ in $D_0.$
3. Let $\mathcal F$ be a family of meromorphic functions in a domain $D.$ Then $\mathcal F$ is normal in $D$ if and only if $\{ \partial(z,f):f \in \mathcal F\}$ is locally uniformly bounded in $D.$ Here $\partial(z,f)$ is the spherical derivative of $f$ at $z.$
My attempt:
Let $\epsilon >0$ be given and let $q \in \mathbb Q.$ Let $\delta = \epsilon.$ Consider
\begin{align}\left|\frac {1}{z-q},\frac {1}{z_0-q}\right| &=|z-q,z_0-q|\\ &=\frac{|z-z_0|}{\sqrt{1+|z-q|^2}\sqrt{1+|z_0-q|^2}}\\ &\leq |z-z_0|\end{align}
It follows that $$|z-z_0|\leq\delta \implies \left|\frac {1}{z-q},\frac {1}{z_0-q}\right|<\epsilon$$ for all $q\in \mathbb Q.$ Thus $\mathcal F$ is equicontinuous with respect to spherical distance and hence normal.
Is the above solution correct? If not, how can I check for normality using above three theorems?
Edit: Another attempt at proving normality:
Let $f_n$ be a sequence in $\mathcal F.$ Then there exists $q_n \in \mathbb Q$ such that $f_n(z)=\frac{1}{z-q_n}.$
Case 1: $q_n \to \infty.$ Then $f_n(z) \to 0$ uniformly.
Case 2: $q_n$ is bounded. Then it has a convergent subsequence $\{q_{n_k}\}.$ Let $r \in \mathbb R$ such that $q_{n_k} \to r.$ Then $f_{n_k} \to \frac{1}{z-r}$ uniformly.
In each case, we get a subsequence which is $C_0.$ Thus, $\mathcal F$ is normal?
Both your solutions are correct. A third option is to verify that the spherical derivative is uniformly bounded: $$ \frac{|f_q'(z)|}{1 + |f_q(z)|^2} = \frac{1}{|z-q|^2 + 1} \le 1 \, . $$