See page 254 of Georg Cantor (1878), "Ein Beitrag zur Mannigfaltigkeitslehre", Journal für die Reine und Angewandte Mathematik, 84: 242–258. This proof is discussed in Joseph Dauben (1979), Georg Cantor: His Mathematics and Philosophy of the Infinite, Harvard University Press, ISBN0-674-34871-0, pp. 61–62, 65. On page 65, Dauben proves a result that is stronger than Cantor's. He lets "φν denote any sequence of rationals in [0, 1]." Cantor lets φν denote a sequence enumerating the rationals in [0, 1], which is the kind of sequence needed for his construction of a bijection between [0, 1] and the irrationals in (0, 1).