Analysis with an Introduction to Proof

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