
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 0132298244

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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 “forwardbackward”
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 researchlevel 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.
