Disk Automorphism and Convergence of Zero

Question

It is understood that for the sequence of disk automorphisms $$\frac{z-a_n}{1-\bar{a}_nz},$$ if $a_n\to a\in D_1$, then this sequence converges uniformly on compact subsets of $D_1$ to the automorphism $$\frac{z-a}{1-\bar{a}z}.$$ I was wondering what we might be able to say about the converse of this statement? Namely, if $$\frac{z-a_n}{1-\bar{a}_nz}\to \frac{z-a}{1-\bar{a}z}$$
on compact subsets of $D_1$, is it necessarily true that $a_n\to a\in D_1$?

Answer 1

If the functions $f_n(z) = \frac{z-a_n}{1-\bar{a}_nz}$ converge to $f(z) = \frac{z-a}{1-\bar{a}z}$ in the unit disk then $$ -a_n = f_n(0) \to f(0) = -a $$ so that necessarily $a_n \to a$.