Typically, undergraduates see real analysis as one of the most diﬃcult courses that a mathematics major is required to take. The main reason for this perception is twofold: Students must comprehend new abstract concepts and learn to deal with these concepts on a level of rigor and proof not previously encountered. A key challenge for an instructor of real analysis is to ﬁnd a way to bridge the gap between a student’s preparation and the mathematical skills that are required to be successful in such a course.
Real Analysis: With Proof Strategies provides a resolution to the "bridging-the-gap problem." The book not only presents the fundamental theorems of real analysis, but also shows the reader how to compose and produce the proofs of these theorems. The detail, rigor, and proof strategies oﬀered in this textbook will be appreciated by all readers.
- Explicitly shows the reader how to produce and compose the proofs of the basic theorems in real analysis
- Suitable for junior or senior undergraduates majoring in mathematics.
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.
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
"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