I am trying to understand the proof that the family of analytic functions with positive real part is normal from here.
One considers the Mobius transformation $T(z)=\frac{1+z}{1-z}$ that maps the right half plane $G$ to the unit disk $D$. Then the family $F'=\{\frac{1+f}{1-f}\,| \, f \in F \}$ being locally bounded is normal.
Thus we get for $g_n \in F' \,\, \exists g_{n_k} \to g $ uniformally on compact subsets of $D$.
How does one show that for $T^{-1}(z)=\frac{z-1}{z+1}$, this implies $T^{-1}(g_{n_k})=f_{n_k} \to f=T^{-1}(g)$ on compact subsets of $G$? Is this true in general or do we need some bound like for all $f \in F$, we have $|f(a)| \leq M$ for some point $a \in G$ where $M >0$.
