A proof of Abel's continuity theorem (Q1066372)

The authors give a new proof of Stolz's form of the Abel continuity theorem: If \(\sum^{\infty}_{n=0}a_ n\) converges and \(f(z)=\sum^{\infty}_{n=0}a_ nz^ n\) then \(\lim_{z\to 1}f(z)=f(1)\), where z is restricted to approach the point 1 in such a way that \(| z| <1\) and \(| 1-z| /(1-| z|)\) remains bounded [cf. \textit{R. Powell} and the reviewer, Summability theory and its applications, 3-6 (1972; Zbl 0248.40001)].