Introduction to Real Analysis, 4th Edition

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  • Edition: 4th
  • Format: Hardcover
  • Copyright: 2011-01-01
  • Publisher: Wiley

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This text provides the fundamental concepts and techniques of real analysis for students in all of these areas. It helps one develop the ability to think deductively, analyse mathematical situations and extend ideas to a new context. Like the first three editions, this edition maintains the same spirit and user-friendly approach with addition examples and expansion on Logical Operations and Set Theory. There is also content revision in the following areas: introducing point-set topology before discussing continuity, including a more thorough discussion of limsup and limimf, covering series directly following sequences, adding coverage of Lebesgue Integral and the construction of the reals, and drawing student attention to possible applications wherever possible.

Table of Contents

Sets and Functions
Mathematical Induction
Finite and Infinite Sets
The Real Numbers
The Algebraic and Order Properties of R
Absolute Value and Real Line
The Completeness Property of R
Applications of the Supremum Property
Sequences and Series
Sequences and Their Limits
Limit Theorems
Monotone Sequences
Subsequences and the Bolzano-Weierstrass Theorem
The Cauchy Criterion
Properly Divergent Sequences
Introduction to Infinite Series
Limits of Functions
Limit Theorems
Some Extensions of the Limit Concept
Continuous Functions
Continuous Functions
Combinations of Continuous Functions
Continuous Functions on Intervals
Uniform Continuity
Continuity and Gauges
Monotone and Inverse Functions
The Derivative
The Mean Value Theorem
L'Hospital's Rules
Taylor's Theorem
The Riemann Integral
The Riemann Integral
Riemann Integrable Functions
The Fundamental Theorem
Approximate Integration
Sequences of Functions
Pointwise and Uniform Convergence
Interchange of Limits
The Exponential and Logarithmic Functions
The Trigonometric Functions
Infinite Series
Absolute Convergence
Tests for Absolute Convergence
Tests for Nonabsolute Convergence
Series of Functions
The Generalized Riemann Integral
Definition and Main Properties
Improper and Lebesgue Integrals
Infinite Intervals
Convergence Theorems
A Glimpse into Topology
Open and Closed Sets in R
Compact Sets
Continuous Functions
Metric Spaces
Logic and Proofs
Finite and Countable Sets
The Riemann And Lebesgue Criteria
Approximate Integration
Two Examples
Photo Credits
Hints for Selected Exercises
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