Understanding Analysis, 2nd Edition Understanding Analysis, 2nd Edition

Understanding Analysis, 2nd Edition

2nd Edition | ISBN: 9781493927128 / 1493927124


expert-verified solutions in this book

Understanding Analysis, 2nd Edition
Buy on Amazon.com

2nd Edition | ISBN: 9781493927128 / 1493927124


expert-verified solutions in this book

Buy on Amazon.com

Table of Contents

Chapter 1

The Real Numbers

1-2 Some Preliminaries Exercises p.11
1-3 The Axiom of Completeness Exercises p.18
1-4 Consequences of Completeness Exercises p.24
1-5 Cardinality Exercises p.29
1-6 Cantor's Theorem Exercises p.32

Chapter 2

Sequences And Series

2-2 The Limit of a Sequences Exercises p.47
2-3 The Algebraic and Order Limit Theorems Exercises p.54
2-4 The Monotone Convergence Theorem and a First Look at Infinite Series Exercises p.59
2-5 Subsequences and the Bolzano-Weierstrass Theorem Exercises p.65
2-6 The Cauchy Criterion Exercises p.70
2-7 Properties of Infinite Series Exercises p.76
2-8 Double Summations and Products of Infinite Series Exercises p.80

Chapter 3

Basic Topology Of R

3-2 Open and Closed Sets Exercises p.93
3-3 Compact Sets Exercises p.99
3-4 Perfect Sets and Connected Sets Exercises p.105
3-5 Baire's Theorem Exercises p.107

Chapter 4

Functional Limits And Continuity

4-2 Functional Limits Exercises p.120
4-3 Continuous Functions Exercises p.126
4-4 Continuous Functions on Compact Sets Exercises p.134
4-5 The Intermediate Value Theorem Exercises p.139
4-6 Sets of Discontinuity Exercises p.141

Chapter 5

The Derivative

5-2 Derivatives and the Intermediate Value Property Exercises p.152
5-3 The Mean Value Theorems Exercises p.160
5-4 A Continuous Nowhere-Differentiable Function Exercises p.163

Chapter 6

Sequences And Series Of Functions

6-2 Uniform Convergence of a Sequence of Functions Exercises p.180
6-3 Uniform Convergence and Differentiation Exercises p.186
6-4 Series of Functions Exercises p.189
6-5 Power Series Exercises p.195
6-6 Taylor Series Exercises p.203
Exercises p.206

Chapter 7

The Riemann Integral

7-2 The Definition of the Riemann Integral Exercises p.222
7-3 Integrating Functions with Discontinuities Exercises p.226
7-4 Properties of the Integral Exercises p.232
7-5 The Fundamental Theorem of Calculus Exercises p.236
7-6 Lebesgue's Criterion for Riemann Integrability Exercises p.239

Chapter 8

Additional Topics

8-1 The Generalized Riemann Integral Exercises p.251
8-2 Metric Spaces and the Baire Category Theorem Exercises p.258
8-3 Euler's Sum Exercises p.265
8-4 Inventing the Factorial Function Exercises p.271
8-5 Fourier Series Exercises p.283
8-6 A Construction of R From Q Exercises p.298
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