Introductory Real Analysis

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  • Edition: 1st
  • Format: Hardcover
  • Copyright: 1999-07-19
  • Publisher: Brooks Cole
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This text for courses in real analysis or advanced calculus is designed specifically to present advanced calculus topics within a framework that will help students more effectively write and analyze proofs. The authors' comprehensive yet accessible presentation for one- or two-term courses offers a balanced depth of topic coverage and mathematical rigor. Instructors will appreciate the flexibility to adjust the material to their individual class requirements. Students will benefit from reinforcing pedagogy often omitted from texts at this level.

Table of Contents

Proofs, Sets, and Functions Proofs Sets Functions Mathematical Induction
The Structure of R Algebraic and Other Properties of R
The Completeness Axiom
The Rational Numbers Are Dense in R Cardinality
Convergence Limit Theorems Subsequences
Monotone Sequences
Bolzano-Weierstrass Theorems Cauchy Sequences
Limits at Infinity Limit Superior and Limit Inferior
Continuous Functions
Continuity and Sequences
Limits of Functions
Consequences of Continuity
Uniform Continuity
Discontinuities and Monotone Functions
The Derivative Mean Value Theorems
Taylor's Theorem
L'Hopital's Rule
Integration Existence of the Riemann
Integral Riemann
Sums Properties of the Riemann
Integral Families of Riemann
Integrable Functions
Fundamental Theorem of Calculus Improper Integrals
Infinite Series
Convergence and Divergence
Absolute and Conditional Convergence
Regrouping and Rearranging Series
Multiplication of Series
Sequences and Series of Functions
Function Sequences Preservation Theorems Series of Functions
Weierstrass Approximation Theorem
Power Series
Convergence Taylor Series
The Riemann-Stieltjes Theorem
Monotone Increasing
Integrators Families of Intergrable
Functions Riemann-Stieltjes Sums
Functions of Bounded Variation
Integrators of Bounded Variations
The Topology of R Open and Closed Sets
Neighborhoods and Accumulation Points Compact Sets
Connected Sets
Continuous Functions
Hints and Answers
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