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Real Analysis: A First Course, 2nd Edition
ISBN-10: 0-201-43727-9
ISBN-13: 978-0-201-43727-0
Published by Addison-Wesley
© 2002
Pub. Date: Jun 1, 2001
Format: Paper
Table of Contents
1. Real Numbers.
What Is a Real Number?
Absolute Value, Intervals, and Inequalities.
The Completeness Axiom.
Countable and Uncountable Sets.
Real-Valued Functions.
2. Sequences.
Convergent Sequences.
Monotone Sequences and Cauchy Sequences.
Subsequences.
Supplementary Exercises.
3. Limits and Continuity.
The Limit of a Function.
Continuous Functions.
Intermediate and Extreme Values.
Uniform Continuity.
Monotone Functions.
Supplementary Exercises.
4. Differentiation.
The Derivative of a Function.
The Mean Value Theorem.
Further Topics on Differentiation.
Supplementary Exercises.
5. Integration.
The Riemann Integral.
Conditions for Riemann Integrability.
The Fundamental Theorem of Calculus.
Further Properties of the Integral.
Numerical Integration.
Supplementary Exercises.
6. Infinite Series.
Convergence of Infinite Series.
The Comparison Tests.
Absolute Convergence.
Rearrangements and Products.
Supplementary Exercises.
7. Sequences and Series of Functions.
Pointwise Convergence.
Uniform Convergence.
Uniform Convergence and Inherited Properties.
Power Series.
Taylor's Formula.
Several Miscellaneous Results.
Supplementary Exercises.
8. Point-Set Topology.
Open and Closed Sets.
Compact Sets.
Continuous Functions.
Miscellaneous Results.
Metric Spaces.
Appendix A. Mathematical Logic.
Mathematical Theories.
Statements and Connectives.
Open Statements and Quantifiers.
Conditional Statements and Quantifiers.
Negation of Quantified Statements.
Sample Proofs.
Some Words of Advice.
Appendix B. Sets and Functions.
Sets.
Functions.
Appendix C. Mathematical Induction.
Three Equivalent Statements.
The Principle of Mathematical Induction.
The Principle of Strong Induction.
The Well-Ordering Property.
Some Comments on Induction Arguments.
Bibliography.
Index.
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