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## ABOUT ** ross elementary analysis 2nd edition solutions pdf **

Undergraduate Texts in Mathematics are generally aimed at third- and fourthyear undergraduate mathematics students at North American universities. These texts strive to provide students and teachers with new perspectives and novel approaches. The books include motivation that guides the reader to an appreciation of interrelations among different aspects of the subject. They feature examples that illustrate key concepts as well as exercises that strengthen understanding.

## Contents

Preface v

1 Introduction 1 1 The Set N of Natural Numbers . . . . . . . . . . . . 1 2 The Set Q of Rational Numbers . . . . . . . . . . . 6 3 The Set R of Real Numbers . . . . . . . . . . . . . 13 4 The Completeness Axiom . . . . . . . . . . . . . . . 20 5 The Symbols +∞ and −∞ . . . . . . . . . . . . . . 28 6 * A Development of R . . . . . . . . . . . . . . . . . 30

2 Sequences 33 7 Limits of Sequences . . . . . . . . . . . . . . . . . . 33 8 A Discussion about Proofs . . . . . . . . . . . . . . 39 9 Limit Theorems for Sequences . . . . . . . . . . . . 45 10 Monotone Sequences and Cauchy Sequences . . . . 56 11 Subsequences . . . . . . . . . . . . . . . . . . . . . . 66 12 limsup’s and liminf’s . . . . . . . . . . . . . . . . . 78 13 * Some Topological Concepts in Metric Spaces . . . 83 14 Series . . . . . . . . . . . . . . . . . . . . . . . . . . 95 15 Alternating Series and Integral Tests . . . . . . . . 105 16 * Decimal Expansions of Real Numbers . . . . . . . 109

Contentsx

3 Continuity 123 17 Continuous Functions . . . . . . . . . . . . . . . . . 123 18 Properties of Continuous Functions . . . . . . . . . 133 19 Uniform Continuity . . . . . . . . . . . . . . . . . . 139 20 Limits of Functions . . . . . . . . . . . . . . . . . . 153 21 * More on Metric Spaces: Continuity . . . . . . . . 164 22 * More on Metric Spaces: Connectedness . . . . . . 178

4 Sequences and Series of Functions 187 23 Power Series . . . . . . . . . . . . . . . . . . . . . . 187 24 Uniform Convergence . . . . . . . . . . . . . . . . . 193 25 More on Uniform Convergence . . . . . . . . . . . . 200 26 Diﬀerentiation and Integration of Power Series . . . 208 27 * Weierstrass’s Approximation Theorem . . . . . . . 216

5 Diﬀerentiation 223 28 Basic Properties of the Derivative . . . . . . . . . . 223 29 The Mean Value Theorem . . . . . . . . . . . . . . 232 30 * L’Hospital’s Rule . . . . . . . . . . . . . . . . . . 241 31 Taylor’s Theorem . . . . . . . . . . . . . . . . . . . 249

6 Integration 269 32 The Riemann Integral . . . . . . . . . . . . . . . . . 269 33 Properties of the Riemann Integral . . . . . . . . . 280 34 Fundamental Theorem of Calculus . . . . . . . . . . 291 35 * Riemann-Stieltjes Integrals . . . . . . . . . . . . . 298 36 * Improper Integrals . . . . . . . . . . . . . . . . . . 331

7 Capstone 339 37 * A Discussion of Exponents and Logarithms . . . . 339 38 * Continuous Nowhere-Diﬀerentiable Functions . . . 347

Appendix on Set Notation 365

Selected Hints and Answers 367

A Guide to the References 394