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Real Analysis


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Table of Contents

1. Proofs, Sets, Functions, and Induction. 1.1. Proofs. 1.2. Sets. 1.3. Functions. 1.4. Mathematical Induction. 2. The Real Numbers. 2.1. Introduction. 2.2. R is an Ordered Field. 2.3 The Completeness Axiom. 2.4. The Archimedean Property. 2.5. Nested Intervals Theorem. 3. Sequences. 3.1 Convergence. 3.2 Limit Theorems for Sequences. 3.3. Subsequences. 3.4. Monotone Sequences. 3.5. Bolzano–Weierstrass Theorems. 3.6. Cauchy Sequences. 3.7. Infinite Limits. 3.8. Limit Superior and Limit Inferior. 4. Continuity. 4.1. Continuous Functions. 4.2. Continuity and Sequences. 4.3. Limits 0f Functions. 4.4. Consequences 0f Continuity. 4.5 Uniform Continuity. 5. Differentiation. 5.1. The Derivative. 5.2. The Mean Value Theorem. 5.3. Taylor’s Theorem. 6. _ Riemann Integration. 6.1. The Riemann Integral. 6.2. Properties of The Riemann Integral. 6.3. Families of Integrable Functions. 6.4. The Fundamental Theorem of Calculus. 7. Infinite Series. 7.1. Convergence and Divergence. 7.2 Convergence Tests. 7.3. Regrouping and Rearranging Terms of a Series. 8. Sequences and Series of Functions. 8.1 Pointwise and Uniform Convergence. 8.2. Preservation Theorems. 8.3. Power Series. 8.4. Taylor Series. Appendix A: Proof of the Composition Theorem. Appendix B: Topology on the Real Numbers. Appendix C: Review of Proof and Logic.

About the Author

Daniel W. Cunningham is a Professor of Mathematics at SUNY Buffalo State, a campus of the State University of New York. He was born and raised in Southern California and holds a Ph.D. in mathematics from the University of California at Los Angeles (UCLA). He is also a member of the Association for Symbolic Logic, the American Mathematical Society, and the Mathematical Association of America.

Cunningham is the author of multiple books. Before arriving at Buffalo State, Professor Cunningham worked as a software engineer in the aerospace industry


"This textbook is intended for undergraduate students who have completed a standard calculus course sequence that covers differentiation and integration and a course that introduces the basics of proof-writing. For students who have a limited proof-writing background, the author includes an abbreviated discussion of proofs, sets, functions, and induction in Chapter 1.

[. . . ] In summary, this book is a good resource for student’s who are taking a first course in real analysis and who have a limited background in proof-writing."
– MAA Reviews"Real Analysis: With Proof Strategies by Professor Daniel W. Cunningham explicitly shows the reader how to produce and compose the proofs of the basic theorems in real analysis and is eminently suitable for junior or senior undergraduates majoring in mathematics."
– Midwest Book Review

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