Introduction to Real Analysis

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.

PrefacePart One: PreliminariesChapter 1. Building ProofsChapter 2. Finite, Infinite, and Even BiggerChapter 3. Algebra of the Real NumbersChapter 4. Ordering, Intervals, and NeighborhoodsPart Two: The Structure of the Real Number SystemChapter 5. Upper Bounds and SupremaChapter 6. Nested IntervalsChapter 7. Cluster PointsChapter 8. Topology of the Real NumbersChapter 9. SequencesChapter 10. Sequences and the Big TheoremChapter 11. Compact SetsChapter 12. Connected SetsPart Three: Topics from CalculusChapter 13. SeriesChapter 14. Uniform ContinuityChapter 15. Sequences and Series of FunctionsChapter 16. DifferentiationChapter 17. IntegrationChapter 18. Interchanging Limit ProcessesPart Four: Selected ShortsChapter 19. Increasing FunctionsChapter 20. Continuous Functions and DifferentiabilityChapter 21. Continuous Functions and IntegrabilityChapter 22. We Build the Real NumbersReferences and further readingIndex