The following question was posed in Greene and Krantz's textbook 3rd edition, page 296 Problem 5:
Let $\{a_n \}$ be a sequence in $\mathbb{D}$, the open unit disk s.t. $\sum_{n \in \mathbb{N}} 1 - |a_n|$ is finite. Let $B$ denote the associated Blaschke product for $\{a_n \}$ defined on $\mathbb{D}$. Let $P \in \partial \mathbb{D}$. Show that the $ \lim_{\mathbb{D} \ni z \to P} D(z)$ exists iff $P$ is not an accumulation point of $\{ a_n : n \in \mathbb{N} \}$.
I have shown that if $P$ is not an accumulation point of $\{ a_n : n \in \mathbb{N} \}$ then $ \lim_{\mathbb{D} \ni z \to P} D(z)$ exists. (Actually then $P$ is regular.) But I haven't been able to figure out the other direction.
BTW, this is not a current homework problem.