Suppose we have a sequence of distinct complex numbers $\{a_n\}$ such that $a_n\rightarrow 1$ and the $a_n$ satisfy the Blaschke condition $\sum (1-|a_n|)<\infty$. Does there exist a Blaschke product $B$ such that $B(a_n)=0$ for each $n$ and $B(1)=1$?
I know that there is a Blaschke product $B$ for which $B(a_n)=0$ for all $n$, and my question is about what control we have over choosing certain boundary values of $B$. Can we ever guarantee that $B(1)$ exists, and if so, can we choose its value?
Thank you for your help.