Alan F. Beardon
This book is intended as an undergraduate text on real analysis and includes all the standard material such as sequences, infinite series, continuity, differentiation, and integration, together with worked examples and exercises. Despite the fact that there are hundreds of books on analysis on the market, by unifying and simplifying all the various notions of limit, the author has successfully presented a unique and novel approach to the subject matter, which has not previously appeared in book form. The author defines what is meant by a limit just once, and all of the subsequent limiting processes will be seen as special cases of this one definition.Accordingly, the subject matter attains a unity and coherence that is missing in the traditional approach. Students will be able to fully appreciate and understand the common source of the topics they are studying while also realizing that they are "variations on a theme" rather than essentially different topics, and therefore, will gain a better understanding of the subject. The book is divided into three sections. Part I contains preliminary material on sets, and on real and complex numbers.Part II starts with the definition of a limit and its basic properties, and continues with three basic results; the Intermediate Value Theorem, the Mean Value inequality, and the Cauchy Criterion, all of which are proved by bisection arguments.
The last chapter in this section contains a detailed discussion of infinite series, including a treatment of unordered sums. Part III comprises the standard material in analysis, and because it follows from the basic ideas presented in the earlier section, much of the material progresses remarkably smoothly.