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Analysis with an Introduction to Proof

Steven R. Lay
Publication Date: 
Number of Pages: 
[Reviewed by
Miklós Bóna
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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


Hints for Selected Exercises