Analysis with an Introduction to Proof

Steven R. Lay  
Total pages
November 2013
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Analysis with an Introduction to Proof


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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.



  • More than 250 true/false questions are unique to this text and tied directly to the narrative; these are perfect for stimulating class discussion and debate.
    • Carefully worded to anticipate common student errors
    • Encourage critical thinking and promote careful reading of the text
    • The justification for a “false" answer is often an example that the students can add to their growing collection of counterexamples.
  • More than 100 practice problems throughout the text provide a simple problem for students to apply what they have just read. Answers are provided just prior to the exercises for reinforcement and for students to check their understanding.
  • Exceptionally high-quality drawings illustrate key ideas.
  • Numerous examples and more than 1,000 exercises give the breadth and depth of practice that students need to learn and master the material.
  • Fill-in-the-blank proofs guide students in the art of writing proofs.
  • Glossary of Key Terms at the end of the book includes 180 key terms with the meaning and page number where each is introduced, providing an invaluable reference when studying, or for future courses.
  • Review of Key Terms after each section emphasizes the importance of definitions and language in mathematics and helps students organize their studying.


New to this Edition

  • Some proofs have been simplified and there are several new examples and illustrations.
  • More than 200 new exercises provide more of the practice that students need to master the material.
  • The definition of a convergent sequence has been changed to include the more traditional requirement that the number limiting the indices should be a natural number (rather than a real number).  This emphasizes the Archimedean property of the real numbers.
  • Animated PowerPoint Presentations are now available for all sections in the text. More than just an outline of each lesson, these were created by the author for use in his own classroom.


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


Hints for Selected Exercises