Resumen de Convergence and the Cauchy Property of Sequences in the Setting of Actual Infinity
B.A. Shipman
Traditional definitions, language, and visualizations of convergence and the Cauchy property of sequences convey a sense of the sequence as a potentially infinite process rather than an actually infinite object. This has a deep-rooted influence on how we think about and teach concepts on sequences, particularly in undergraduate calculus and analysis. After characterizing this point of view, this paper reformulates the definitions of convergence and the Cauchy property in the setting of actual infinity. This yields a conceptually streamlined approach and simple proofs of classic results on sequences. The paper also presents pedagogical metaphors that guide students in defining limit and the Cauchy property from the actually infinite standpoint.
