I am having problem with showing that if sequence of finite Blaschke Products
$$B_n=z^m\prod_{k=1}^{n}\frac{|z_k|}{z_k}\frac{z_k - z}{1 - \overline{z}_k z}$$
where $(z_n)_{n\in \mathbb{N}} \subset \mathbb{D}$ and $|z_n| \rightarrow 1$ as $n\rightarrow \infty$, converges uniformly to $B$ on compact subsets of open unit disk, then function sequence $(\frac{B_n(z)}{z^m})$ converges uniformly to $\frac{B}{z^m}$ on compact subsets of open unit disk.
I got stuck to this statement. Hint for this is that Maximum principle should be used but don't know how to apply it.