Analysis: With an Introduction to Proof

About the Book

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.

Table of Contents:
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 :
"Let me begin by saying that I really like this book, and I do not say that of very many books. What impresses me most is the level of motivation and explanation given for the basic logic, the construction of proofs, and the ways of thinking about proofs that this book provides in its first few sections. It felt that the author was talking to the reader the way I would like to talk to students. There was an air of familiarity there. All kinds of useful remarks were made, the type I would like to make in my lectures." -- Aimo Hinkkanen, University of Illinois at Urbana "The writing style is suitable for our students. It is clear, logical, and concise. The examples are very helpful and well-developed. The topics are thoroughly covered and at the appropriate level for our students. The material is technically accurate, and the pedagogical material is effectively presented." -- John Konvalina, University of Nebraska at Omaha