Abstract
We have seen that the beginning of the theory of limits, of functions and of sequences, as presented in the last two sections, does not require any assumption that ℝ “has no holes in it”; if, instead of ℝ, we deal with the rational number system [Q, +, •, <], then the theorems in Sections 3 and 4 still hold, and their proofs are exactly the same, because the rational numbers form an ordered field. On this basis, the familiar derivations of the elementary differentiation formulas become complete proofs; all that was needed, to justify them, was a valid definition of a limit, and simple theorems based on it.
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© 1982 Springer-Verlag New York, Inc.
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Moise, E.E. (1982). The Continuity of IR. In: Introductory Problem Courses in Analysis and Topology. Universitext. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-8183-9_5
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DOI: https://doi.org/10.1007/978-1-4613-8183-9_5
Publisher Name: Springer, New York, NY
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