This is a textbook for a first course in real analysis for undergraduates. It is a gentle introduction which, for example, points out various logically equivalent ways of saying “P implies Q” but yet without treating the metamathematics. The first 11 chapters include preliminary material and a quite standard treatment of limits, continuity and differentiation in \({\mathbb{R}}\) and \({\mathbb{R}}^ n\), and associated topics, based on the definition of \({\mathbb{R}}\) as a completely ordered field. (Curiously, the fact that the natural numbers are well ordered, stated immediately after the induction principle, appears to be an independent assumption rather than a consequence of the latter). A section on optimization includes treatment of the Kuhn-Tucker conditions. Chapters 12 through 18 are devoted to a unified treatment of integration theory via the gauge integral, which is more in the spirit of the Riemann integral, yet has the nice convergence properties of the Lebesgue integral. (It is actually equivalent to the Perron integral, and more general than that of Lebesgue.) The last 12 chapters are on analysis in metric spaces, and include such topics as contraction maps, the Baire category, Arzelà-Ascoli and Stone-Weierstrass theorems, and Fréchet derivatives. There is a surprizing amount of coverage for a book of under 350 pages which really does seem to be a gentle introduction to real analysis. An ample supply of (800) exercises include a good mixture of illustrative examples, some of the more routine proofs of text propositions, and extensions of the theory. This book is an excellent text at its intended level.