Preface

Mathematical analysis is central to mathematics curricula not only because it is a stepping-stone to the study of advanced analysis, but also because of its applications to other branches of mathematics, physics, and engineering at both the undergraduate and graduate levels. Although there are many texts on this subject under various titles such as “Analysis,” “Advanced Calculus,” and “Real Analysis,” there seems to be a need for a text that explains fundamental concepts with motivating examples and with a geometric flavor wherever it is appropriate. It is hoped that this book will serve that need. This book provides an introduction to mathematical analysis for students who have some familiarity with the real number system. Many ideas are explained in more than one way with accompanying figures in order to help students to think about concepts and ideas in several ways. It is hoped that through this book, both student and teacher will enjoy the beauty of some of the arguments that are often used to prove key theorems—regardless of whether the proofs are short or long. The distinguishing features of the book are as follows. It gives a largely self-contained and rigorous introduction to mathematical analysis that prepares the student for more advanced courses by making the subject matter interesting and meaningful. The exposition of standard material has been done with extra care and abundant motivation. Unlike many standard texts, the emphasis in the present book is on teaching these topics rather than merely presenting the standard material. The book is developed through patient explanations, motivating examples, and pictorial illustrations conveying geometric intuition in a pleasant and informal style to help the reader grasp difficult concepts easily. Each section ends with a carefully selected set of “Questions” and “Exercises.” The questions are intended to stimulate the reader to think, for example, about the nature of a definition or the fate of a theorem without one or more of its hypotheses. The exercises cover a broad spectrum of difficulty and are intended not only for routine problem solving, but also to deepen