## Introduction to Real Analysis, 1/e

Michael J. Schramm, Le Moyne College

Published August, 1995 by Prentice Hall Engineering/Science/Mathematics

Copyright 1996, 368 pp.
Cloth
ISBN 0-13-229824-4

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Real Analysis-Mathematics

Advanced Calculus-Mathematics

This beautifully written text starts with proofs and sets in the first 40 pages and continues in the rest of Parts I and II to maintain an ongoing emphasis on the construction of proofs, demonstrating proper skills through detailed examples using the “forward-backward” method.

one of the text's greatest strengths are the problem sets, which are many and varied. In addition to a large number of more traditional problems, students are asked to complete partial proofs, find flaws in incorrect proofs, and modify proofs in the light of new information.
offers a wide range of problem material—many of which follow a “stream of consciousness” format—guiding readers through large projects and allowing them to explore and develop interest in near research-level topics.
presents the most abstract subject matter in terms that relate to students' experience in calculus, rather than ignoring or downplaying the value of this experience.
depicts the structure of the real number system as a collection of closely interrelated properties, rather than simply a list of theorems.
covers a wide variety of topics in Part IV, each explored using a “discovery” process.
conveys concepts in an interesting, conversational tone, presenting the subject as one that is open and appealing to everyone.

I. PRELIMINARIES.

1. Building Proofs.
2. Finite, Infinite and Even Bigger.
3. Algebra of Real Numbers.
4. Ordering, Intervals and Neighborhoods.

II. THE STRUCTURE OF THE REAL NUMBER SYSTEM.
5. Upper Bounds and Suprema.
6. Nested Intervals.
7. Cluster Points.
8. Topology of the Real Numbers.
9. Sequences.
10. Sequences and the Big Theorem.
11. Compact Sets.
12. Connected Sets.
III. TOPICS FROM CALCULUS.
13. Series.
14. Uniform Continuity.
15. Sequences and Series of Functions.
16. Differentiation.
17. Integration.
18. Interchanging Limit Processes.
IV. SELECTED SHORTS.
19. Increasing Functions.
20. Continuous Functions and Differentiability.
21. Continuous Functions and Integrability.
22. We Build the Real Numbers.
Index.

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