## Table of Contents

**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