Calculus and the Real Numbers

Abstract

Section 1 establishes some conventions about sets and functions. The next three sections are devoted to constructing the real numbers as certain Cauchy sequences of rational numbers, and investigating their order and arithmetic. The rest of the chapter deals with the basic ideas of the calculus of one variable. Topics covered include continuity, the convergence of sequences and series of continuous functions, differentiation, integration, Taylor’s theorem, and the basic properties of the exponential and trigonometric functions and their inverses. Most of the material is a routine constructivization of the corresponding part of classical mathematics; for this reason it affords a good introduction to the constructive approach.