(from preface)

This book is written primarily for undergraduates, though Part I may be judged by some teachers to be suitable for mathematicians during their last year at school. It includes the convergence theory that is commonly required for a university honours course in pure and applied mathematics, but excludes topics appropriate to post-graduate or to highly specialized courses of study. It has taken shape from sets of lectures I have given at various times during some fifteen years of university teaching.

The book develops the theory of convergence on the basis of two fundamental assumptions (one about upper bounds, one about irrational number as the limit of a sequence of rational numbers). With these assumptions the theory of convergence can be developed without appeal to the properties of Dedekind cuts. The 'real number' appears in the appendix, where the assumptions of the book are proved to be consequences of the definition of 'real number'.