The Real Numbers and Real Analysis

This rigorous, detailed introduction to real analysis presents the fundamentals clearly and includes definitions, theorems and proofs. Mirroring the structure of standard calculus courses makes it especially accessible to university students of mathematics.

This text is a rigorous, detailed introduction to real analysis that presents the fundamentals with clear exposition and carefully written definitions, theorems, and proofs. It is organized in a distinctive, flexible way that would make 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 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 calculus.

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 making use of sequences for proofs Presents the equivalence of various important theorems of real analysis with the Least Upper Bound Property Relates real analysis to previously learned materal, including detailed discussion of such topics as the transcendental functions, area and the number pi Offers three different entryways into the study of real numbers, depending on the student audience Contains historical context, biographical anecdotes, and reflections on the material in each chapter Includes over 350 exercises, reinforcing concepts Includes supplementary material: sn.pub/extras

Autorentext
Dr. Ethan D. Bloch of Bard College is the author of two Springer publications "A First Course in Geometric Topology and Differential Geometry," and the first and second editions of, "Proofs and Fundamentals: A First Course in Abstract Mathematics." More information about Dr. Ethan D. Bloch can be found on his person web page: http://math.bard.edu/bloch

Zusammenfassung
The Real Numbers and Real Analysis 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 calculus.

Inhalt
Preface.-To the Student.-To the Instructor.- 1. Construction of the Real Numbers.- 2. Properties of the Real Numbers.- 3. Limits and Continuity.- 4. Differentiation.- 5. Integration.- 6. Limits to Infinity.-7. Transcental Functions.-8. Sequences.- 9. Series.- 10. Sequences and Series of Functions.- Bibliography.- Index.