A Lindelöf Property for Uniformly Normal Families

Abstract

In this note we present a simple new proof for a Lindelöf property for a normal map accessible to advanced undergraduate students. The proof extends the result to uniformly normal families: If $\{ f_n :D \to P^1(\mathbb{C})\}$ is a uniformly normal sequence from the unit disk $D$ in the complex plane ${\bf \mathbb{C}}$ into the Riemann Sphere $P^1(\mathbb{C})$ such that $\lim _{r_n \to 1}f_n(r_n)$ exists for all $ \{r_n\}\subset (0,1),$ then the sequence $\{f_n\}$ has non-tangential limit at $1$.