This book 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.
This book 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.
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
Inhaltsangabe
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.
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.
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