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Elementary Analysis: The Theory of Calculus

Table of Contents

Preface

Introduction
1 The Set N of Natural Numbers
2 The Set Q of Rational Numbers
3 The Set R of Real Numbers
4 The Completeness Axiom
5 The Symbols +∞ and –∞
6 *A Development of R

Sequences
7 Limits of Sequences
8 A Discussion about Proofs
9 Limit Theorems for Sequences
10 Monotone Sequences and Cauchy Sequences
11 Subsequences
12 lim sup's and lim inf's
13 *Some Topological Concepts in Metric Spaces
14 Series
15 Alternating Series and Integral Tests
16 *Decimal Expansions of Real Numbers

Continuity
17 Continuous Functions
18 Properties of Continuous Functions
19 Uniform Continuity
20 Limits of Functions
21 *More on Metric Spaces: Continuity
22 *More on Metric Spaces: Connectedness

Sequences and Series of Functions
23 Power Series
24 Uniform Convergence
25 More on Uniform Convergence
26 Differentiation and Integration of Power Series
27 *Weierstrass's Approximation Theorem

Differentiation
28 Basic Properties of the Derivative
29 The Mean Value Theorem
30 *L'Hospital's Rule
31 Taylor's Theorem

Integration
32 The Riemann Integral
33 Properties of the Riemann Integral
34 Fundamental Theorem of Calculus
35 *Riemann-Stieltjes Integrals
36 * Improper Integrals
37 *A Discussion of Exponents and Logarithms

Appendix on Set Notation

Selected Hints and Answers

References

Symbols Index

Index

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