Acutely aware of the need for rigor, the student is much better prepared to understand what constitutes a proper mathematical proof and how to write one.Fifteen years of classroom experience with the first edition of Understanding Analysis have solidified and refined the central narrative of the second edition.

Chapter 3, on in nite combinatorics and graphs, may also be studied independently, although many readers will want to investigate trees, matchings, and Ramsey theory for nite sets before exploring these topics for in nite sets in the third chapter.

Mathematics is playing an ever more important role in the physical and biological sciences, provoking a blurring of boundaries between scientific disciplines and a resurgence of interest in the modern as well as the clas­ sical techniques of applied mathematics.