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The book is somewhat unusual in bringing together its two main subjects, usually addressed in separate courses. The first of these subjects is an introduction to the notion of proofs and proof methods, and the second one is basic analysis.
The first two chapters, Logic and Proof and Sets and Functions, are typically covered in a “Transition to Higher Mathematics” class. They are usually discussed in in books whose goal is either simply to teach students to prove statements, or to teach students how to prove statements and introduce them to discrete mathematics. In this book, however, the chapters that follow (Real Numbers, Sequences, Limits) slowly lead to Analysis, at the level of precision that is normally seen in Advanced Calculus classes taken by upper-level undergraduates.
There are enough exercises, and a little bit less than half of them have their answers included in the book, sometimes with a little bit of explanation.
The book fills an existing gap by matching these two topics, proofs and analysis, which are usually not taught in the same course. This is great if your students need such a class, but in many places, it can create overlaps — if most students in the class already took a “Transition to Higher Mathematics” class, then they will not need the first two chapters of the book. Another concern is that if we teach students how to prove statements, and all statements we prove in the next three months come from Analysis, then we might send the message that proofs are something used only in Analysis. If you can allay these concerns, then the book may be a good choice for your class.
Miklós Bóna is Professor of Mathematics at the University of Florida.
1. Logic and Proof
Section 1. Logical Connectives
Section 2. Quantifiers
Section 3. Techniques of Proof: I
Section 4. Techniques of Proof: II
2. Sets and Functions
Section 5. Basic Set Operations
Section 6. Relations
Section 7. Functions
Section 8. Cardinality
Section 9. Axioms for Set Theory(Optional)
3. The Real Numbers
Section 10. Natural Numbers and Induction
Section 11. Ordered Fields
Section 12. The Completeness Axiom
Section 13. Topology of the Reals
Section 14. Compact Sets
Section 15. Metric Spaces (Optional)
4. Sequences
Section 16. Convergence
Section 17. Limit Theorems
Section 18. Monotone Sequences and Cauchy Sequences
Section 19. Subsequences
5. Limits and Continuity
Section 20. Limits of Functions
Section 21. Continuous Functions
Section 22. Properties of Continuous Functions
Section 23. Uniform Continuity
Section 24. Continuity in Metric Space (Optional)
6. Differentiation
Section 25. The Derivative
Section 26. The Mean Value Theorem
Section 27. L'Hospital's Rule
Section 28. Taylor's Theorem
7. Integration
Section 29. The Riemann Integral
Section 30. Properties of the Riemann Integral
Section 31. The Fundamental Theorem of Calculus
8. Infinite Series
Section 32. Convergence of Infinite Series
Section 33. Convergence Tests
Section 34. Power Series
9. Sequences and Series of Functions
Section 35. Pointwise and uniform Convergence
Section 36. Application of Uniform Convergence
Section 37. Uniform Convergence of Power Series
Glossary of Key Terms
References
Hints for Selected Exercises
Index