Introduction to Real Analysis

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Overview


This text forms a bridge between courses in calculus and real analysis. It focuses on the construction of mathematical proofs as well as their final content. Suitable for upper-level undergraduates and graduate students of real analysis, it also provides a vital reference book for advanced courses in mathematics.The four-part treatment begins with an introduction to basic logical structures and techniques of proof, including discussions of the cardinality concept and the algebraic and order structures of the real and rational number systems. Part Two presents in-depth examinations of the completeness of the real number system and its topological structure. Part Three reviews and extends the previous explorations of the real number system, and the final part features a selection of topics in real function theory. Numerous and varied exercises range from articulating the steps omitted from examples and observing mechanical results at work to the completion of partial proofs within the text.
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Product Details

  • ISBN-13: 9780486469133
  • Publisher: Dover Publications
  • Publication date: 11/24/2008
  • Series: Dover Books on Mathematics Series
  • Pages: 384
  • Sales rank: 996,020
  • Product dimensions: 5.30 (w) x 8.40 (h) x 0.90 (d)

Meet the Author


Michael J. Schramm has a Ph.D. from Syracuse University.
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Table of Contents


Preface
Part One: Preliminaries
Chapter 1. Building Proofs
Chapter 2. Finite, Infinite, and Even Bigger
Chapter 3. Algebra of the Real Numbers
Chapter 4. Ordering, Intervals, and Neighborhoods
Part Two: The Structure of the Real Number System
Chapter 5. Upper Bounds and Suprema
Chapter 6. Nested Intervals
Chapter 7. Cluster Points
Chapter 8. Topology of the Real Numbers
Chapter 9. Sequences
Chapter 10. Sequences and the Big Theorem
Chapter 11. Compact Sets
Chapter 12. Connected Sets
Part Three: Topics from Calculus
Chapter 13. Series
Chapter 14. Uniform Continuity
Chapter 15. Sequences and Series of Functions
Chapter 16. Differentiation
Chapter 17. Integration
Chapter 18. Interchanging Limit Processes
Part Four: Selected Shorts
Chapter 19. Increasing Functions
Chapter 20. Continuous Functions and Differentiability
Chapter 21. Continuous Functions and Integrability
Chapter 22. We Build the Real Numbers
References and further reading
Index
 
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