Analysis with an Introduction to Proof

Steven Lay is a Professor of Mathematics at Lee University in Cleveland, TN.  He received M.A. and Ph.D. degrees in mathematics from the University of California at Los Angeles.  He has authored three books for college students, from a senior level text on Convex Sets to an Elementary Algebra text for underprepared students.  The latter book introduced a number of new approaches to preparing students for algebra and led to a series of books for middle school math.  Professor Lay has a passion for teaching, and the desire to communicate mathematical ideas more clearly has been the driving force behind his writing.  He comes from a family of mathematicians, with his father Clark Lay having been a member of the School Mathematics Study Group in the 1960s and his brother David Lay authoring a popular text on Linear Algebra. He is a member of the American Mathematical Society, the Mathematical Association of America, and the Association of Christians in the Mathematical Sciences.

Table of Contents

Logic and Proof

Section 1. Logical Connectives
Section 2. Quantifiers
Section 3. Techniques of Proof: I
Section 4. Techniques of Proof: II


Sets and Functions

Section 5. Basic Set Operations
Section 6. Relations
Section 7. Functions
Section 8. Cardinality
Section 9. Axioms for Set Theory(Optional)


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)


Sequences

Section 16. Convergence
Section 17. Limit Theorems
Section 18. Monotone Sequences and Cauchy Sequences
Section 19. Subsequences


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)


Differentiation

Section 25. The Derivative
Section 26. The Mean Value Theorem
Section 27. L’Hôpital’s Rule
Section 28. Taylor’s Theorem


Integration

Section 29. The Riemann Integral
Section 30. Properties of the Riemann Integral
Section 31. The Fundamental Theorem of Calculus


Infinite Series

Section 32. Convergence of Infinite Series
Section 33. Convergence Tests
Section 34. Power Series


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