This is an informal look at the nature of the real numbers, including their construction and their most important analytic and topological properties. Most things are sketched and motivated by geometric arguments, rather than proven rigorously. The book’s goal is not merely to construct the real numbers, but also to show why a good understanding of them is useful. There are extensive historical notes about the evolution of real analysis and our understanding of real numbers.
The book begins with the integers and the rational numbers. It takes these as given, although it does introduce some of the Peano postulates and shows how they can be used to define the arithmetic operations. The construction of the real numbers is then worked out in considerable detail, using Dedekind cuts. The book then continues with Riemann integrals, open and closed sets and the Bolzano–Weierstrass theorem, measure (through Borel sets) and a little bit of Lebesgue integration.
There are two chapters in the middle of the book, on transfinite ordinals and on the axiom of choice, that don’t fit well. They’re well done, but are introduced without explanation of where we are going and in fact it appears are only needed for the construction of non-measurable functions. They seem to be overkill.
The competition in this field is Landau’s Foundations of Analysis. Stillwell has deliberately set out to provide a different sort of construction where you understand what the foundation is supporting and why it is important. I think this is very successful, and his book, although much less rigorous that Landau’s, is much more informative and enjoyable.
Allen Stenger is a math hobbyist and retired software developer. He is webmaster and newsletter editor for the MAA Southwestern Section and is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis. He volunteers in his spare time at MathNerds.org, a math help site that fosters inquiry learning.
The Fundamental Questions
From Discrete to Continuous
Functions and Limits
Open Sets and Continuity
The Axiom of Choice