The real numbers and real analysis - Bloch, Ethan D. 1956- (Springer New York, 2011)
This text is a rigorous, detailed introduction to real analysis that presents the fundamentals with clear exposition and carefully written definitions, theorems, and proofs. The choice of material and the flexible organization, including three different entryways into the study of the real numbers, making it equally appropriate to undergraduate mathematics majors who want to continue in mathematics, and to future mathematics teachers who want to understand the theory behind calculus. The Real Numbers and Real Analysis is accessible to students who have prior experience with mathematical proofs and who have not previously studied real analysis. The text includes over 350 exercises. Key features of this textbook: - provides an unusually thorough treatment of the real numbers, emphasizing their importance as the basis of real analysis - presents material in an order resembling that of standard calculus courses, for the sake of student familiarity, and for helping future teachers use real analysis to better understand calculus - emphasizes the direct role of the Least Upper Bound Property in the study of limits, derivatives and integrals, rather than relying upon sequences for proofs; presents the equivalence of various important theorems of real analysis with the Least Upper Bound Property - includes a thorough discussion of some topics, such as decimal expansion of real numbers, transcendental functions, area and the number p, that relate to calculus but that are not always treated in detail in real analysis texts - offers substantial historical material in each chapter. This book will serve as an excellent one-semester text for undergraduates majoring in mathematics, and for students in mathematics education who want a thorough understanding of the theory behind the real number system and calcul