Analysis with an Introduction to Proof
Description
For courses in undergraduate Analysis and Transition to Advanced Mathematics.
Analysis with an Introduction to Proof, Fifth Edition helps fill in the groundwork students need to succeed in real analysis-often considered the most difficult course in the undergraduate curriculum. By introducing logic and emphasizing the structure and nature of the arguments used, this text helps students move carefully from computationally oriented courses to abstract mathematics with its emphasis on proofs. Clear expositions and examples, helpful practice problems, numerous drawings, and selected hints/answers make this text readable, student-oriented, and teacher- friendly.
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Content
- 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