An eigenvector proof of Fatou's lemma for continuous functions (Q1908703)

The author gives a new first-course-in-analysis proof of the bounded convergence theorem for continuous functions on \([0, 1]\). The proof depends on an ``additive diagonal lemma''. The author also proves that this lemma leads to the ``sup - lim sup theorem'' for Banach spaces. Then he shows how this theorem gives a result on weak convergence in Banach spaces that generalizes a theorem of \textit{J. Rainwater} [Proc. Am. Math. Soc. 14, 999 (1963; Zbl 0117.08302)].