Analysis: With an Introduction to Proof (4e)

For courses in undergraduate Analysis (an easy one) and Transition to Advanced Mathematics.

This text 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 by emphasizing the structure and nature of the arguments used, Lay 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 and student-oriented, and teacher- friendly.

Chapter 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.

• Review of Key Terms after each section—Emphasizes the important role of definitions and language in mathematics and helps students organize their studying.

• Glossary of Key Terms at the end of the book—Includes 180 key terms with the meaning and page number where each is introduced; a great help to students.

• 6 New Figures and more than 150 new exercises—Illustrate key ideas and encourage student practice.

• Clarification of exposition in several sections. cf. Example 16.3, Theorem 21.14, Corollary 21.15, Example 21.16, Theorem 22.2, and Theorem 27.2.

• NEW - Review of Key Terms after each section—Emphasizes the important role of definitions and language in mathematics and helps students organize their studying.

• NEW - Glossary of Key Terms at the end of the book—Includes 180 key terms with the meaning and page number where each is introduced; a great help to students.

• NEW - 6 New Figures and more than 150 new exercises—Illustrate key ideas and encourage student practice.

• NEW - Clarification of exposition in several sections. cf. Example 16.3, Theorem 21.14, Corollary 21.15, Example 21.16, Theorem 22.2, and Theorem 27.2.

• More than 250 true/false questions—Unique to this text and tied directly to the narrative; good for starting class discussion and debate.

  • Carefully worded to anticipate common student errors.

  • Encourage students to read the text carefully and think critically about what they have read.

  • Often the justification for an answer of "false" will be an example that the students can add to their growing collection of counterexamples.

• More than 100 practice problems throughout the text—A key strength of this text.

  • Lets students work a simple problem relating to what they have just read.

  • Provides answers at the end of each section just prior to the exercises for immediate reinforcement and a check of student understanding.

• Exceptionally high-quality drawings to illustrate key ideas.

• Numerous examples and more than 1,000 exercises.

• Fill-in-the-blank proofs—Guides students in the art of writing proofs.