9780130898791

Analysis with an Introduction to Proof

by
  • ISBN13:

    9780130898791

  • ISBN10:

    0130898791

  • Edition: 3rd
  • Format: Hardcover
  • Copyright: 1/1/2001
  • Publisher: Pearson College Div
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Summary

For courses in Real Analysis, Advanced Calculus, and Transition to Advanced Mathematics or Proofs course. Carefully focused on reading and writing proofs, this introduction to the analysis of functions of a single real variable helps students in the transition from computationally oriented courses to abstract mathematics by its emphasis on proofs. Student oriented and instructor friendly, it features clear expositions and examples, helpful practice problems, many drawings that illustrate key ideas, and hints/answers for selected exercises.

Table of Contents

Preface ix
Logic and Proof
1(30)
Logical Connectives
1(9)
Quantifiers
10(5)
Techniques of Proof: I
15(8)
Techniques of Proof: II
23(8)
Sets and Functions
31(56)
Basic Set Operations
31(13)
Relations
44(9)
Functions
53(14)
Cardinality
67(12)
Axioms for Set Theory
79(8)
The Real Numbers
87(52)
Natural Numbers and Induction
87(8)
Ordered Fields
95(9)
The Completeness Axiom
104(11)
Topology of the Reals
115(8)
Compact Sets
123(6)
Metric Spaces
129(10)
Sequences
139(32)
Convergence
139(9)
Limit Theorems
148(8)
Monotone Sequences and Cauchy Sequences
156(6)
Subsequences
162(9)
Limits and Continuity
171(39)
Limits of Functions
171(9)
Continuous Functions
180(8)
Properties of Continuous Functions
188(6)
Uniform Continuity
194(7)
Continuity in Metric Spaces
201(9)
Differentiation
210(33)
The Derivative
210(9)
The Mean Value Theorem
219(8)
I'Hospital's Rule
227(8)
Taylor's Theorem
235(8)
Integration
243(24)
The Riemann Integral
243(8)
Properties of the Riemann Integral
251(8)
The Fundamental Theorem of Calculus
259(8)
Infinite Series
267(24)
Convergence of Infinite Series
267(8)
Convergence Tests
275(10)
Power Series
285(6)
Sequences and Series of Functions
291(27)
Pointwise and Uniform Convergence
291(9)
Applications of Uniform Convergence
300(9)
Uniform Convergence of Power Series
309(9)
References 318(1)
Hints for Selected Exercises 319(18)
Index 337

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