For over three decades, this best-selling classic has been used by thousands of students in the United States and abroad as a must-have textbook for a transitional course from calculus to analysis. It has proven to be very useful for mathematics majors who have no previous experience with rigorous proofs. Its friendly style unlocks the mystery of writing proofs, while carefully examining the theoretical basis for calculus. Proofs are given in full, and the large number of well-chosen examples and exercises range from routine to challenging. The second edition preserves the book’s clear and concise style, illuminating discussions, and simple, well-motivated proofs. New topics include material on the irrationality of pi, the Baire category theorem, Newton's method and the secant method, and continuous nowhere-differentiable functions. Review from the first edition: "This book is intended for the student who has a good, but naïve, understanding of elementary calculus and now wishes to gain a thorough understanding of a few basic concepts in analysis.... The author has tried to write in an informal but precise style, stressing motivation and methods of proof, and ... has succeeded admirably." —MATHEMATICAL REVIEWS
Table of Contents
Cover
Elementary Analysis - The Theory of Calculus, Second Edition
ISBN 9781461462705 ISBN 9781461462712
Preface
Contents
Introduction
�1 The Set N of Natural Numbers
�2 The Set Q of Rational Numbers
2.1 Definiion.
2.2 Rational Zeros Theorem.
2.3 Corollary.
2.4 Remark.
�3 The Set R of Real Numbers
3.1 Theorem.
3.2 Theorem.
3.3 Definiion.
3.4 Definiion.
3.5 Theorem. (i) |a|=0 for all a R. (ii) |ab| = |a|�|b| for.all a, b R. (iii) |a + b|=|a| + |b| for all .a, b . R. Proof (i) is
3.6 Corollary.
3.7 Triangle Inequality.
�4 The Completeness Axiom
4.1 Definiion.
4.2 Definiion.
4.3 Definiion.
4.4 Completeness Axiom.
4.5 Corollary.
4.6 Archimedean Property.
4.7 Denseness of Q.
�5 The Symbols +8 and -8
�6 * A Development of R
Sequences
�7 Limits of Sequences
7.1 Definiion.
�8 A Discussion about Proofs
�9 Limit Theorems for Sequences
9.1 Theorem.
9.2 Theorem.
9.3 Theorem.
9.4 Theorem.
9.5 Lemma.
9.6 Theorem.
9.7 Theorem (Basic Examples). (a) lim 1
9.8 Definiion.
9.9 Theorem.
9.10 Theorem.
�10 Monotone Sequences and Cauchy Sequences
10.1 Definiion.
10.2 Theorem.
10.3 Discussion of Decimals.
10.4 Theorem.
10.5 Corollary.
10.6 Definiion.
10.7 Theorem.
10.8 Definiion.
10.9 Lemma.
10.10 Lemma.
10.11 Theorem.
�11 Subsequences
11.1 Definiion.
11.2 Theorem.
11.3 Theorem.
11.4 Theorem.
11.5 Bolzano-Weierstrass Theorem.
11.6 Definiion.
11.7 Theorem.
11.8 Theorem.
11.9 Theorem.
�12 lim sup's and lim inf's
12.1 Theorem.
12.2 Theorem.
12.3 Corollary.
�13 * Some Topological Concepts in Metric Spaces
13.1 Definiion.
13.2 Definiion.
13.3 Lemma.
13.4 Theorem.
13.5 Bolzano-Weierstrass Theorem.
13.6 Definiion.
13.7 Discussion.
13.8 Definiion.
13.9 Proposition.
13.10 Theorem.
13.11 Definiion.
13.12 Heine-Borel Theorem.
13.13 Proposition.
�14 Series
14.1 Summation Notation.
14.2 Infinit Series.
14.3 Definiion.
14.4 Theorem.
14.5 Corollary.
14.6 Comparison Test.
14.7 Corollary.
14.8 Ratio Test.
14.9 Root Test.
14.10 Remark.
�15 Alternating Series and Integral Tests
15.1 Theorem.
15.2 Integral Tests.
15.3 Alternating Series Theorem.
�16 * Decimal Expansions of Real Numbers
16.1 Long Division.
16.2 Theorem.
16.3 Theorem.
16.4 Definiion.
16.5 Theorem.
Continuity
�17 Continuous Functions
17.1 Definiion.
17.2 Theorem.
17.3 Theorem.
17.4 Theorem.
17.5 Theorem.
�18 Properties of Continuous Functions
18.1 Theorem.
18.2 Intermediate Value Theorem.
18.3 Corollary.
18.4 Theorem.
18.5 Theorem.
18.6 Theorem.
�19 Uniform Continuity
19.1 Definiion.
19.2 Theorem.
19.3 Discussion.
19.4 Theorem.
19.5 Theorem.
19.6 Theorem.
�20 Limits of Functions
20.1 Definiion.
20.2 Remarks. (a) From De.nition 17.1 we see that a function f is continuous at
20.3 Definiion. (a) For a . R and a function f we write limx a f (x)=L provided
20.4 Theorem.
20.5 Theorem.
20.6 Theorem.
20.7 Corollary.
20.8 Corollary.
20.9 Discussion.
20.10 Theorem.
20.11 Remark.
�21 * More on Metric Spaces: Continuity
21.1 Definiion.
21.2 Proposition.
21.3 Theorem.
21.4 Theorem.
21.5 Corollary.
21.6 Remark.
21.7 Theorem.
21.8 Baire Category Theorem.
21.9 Corollary.
21.10 Discussion.
21.11 Theorem.
�22 * More on Metric Spaces: Connectedness
22.1 Definiion.
22.2 Theorem.
22.3 Corollary.
22.4 Definiion.
22.5 Theorem.
0)
Author(s): Kenneth A. Ross
Series: Undergraduate Texts in Mathematics
Edition: 2nd ed. 2013
Publisher: Springer
Year: 2013
Language: English
Pages: 422
Cover......Page 1
Elementary Analysis - The Theory of Calculus, Second Edition......Page 4
ISBN 9781461462705 ISBN 9781461462712......Page 5
Preface......Page 6
Contents......Page 10
§1 The Set N of Natural Numbers......Page 14
§2 The Set Q of Rational Numbers......Page 19
2.1 Definiion.......Page 21
2.2 Rational Zeros Theorem.......Page 22
2.3 Corollary.......Page 23
2.4 Remark.......Page 25
§3 The Set R of Real Numbers......Page 26
3.1 Theorem.......Page 28
3.2 Theorem.......Page 29
3.5 Theorem. (i) |a|=0 for all a R. (ii) |ab| = |a|·|b| for.all a, b R. (iii) |a + b|=|a| + |b| for all .a, b . R. Proof (i) is......Page 30
3.7 Triangle Inequality.......Page 31
4.1 Definiion.......Page 33
4.2 Definiion.......Page 34
4.3 Definiion.......Page 35
4.5 Corollary.......Page 36
4.7 Denseness of Q.......Page 38
§5 The Symbols +8 and -8......Page 41
§6 * A Development of R......Page 43
§7 Limits of Sequences......Page 46
7.1 Definiion.......Page 48
§8 A Discussion about Proofs......Page 52
9.1 Theorem.......Page 58
9.4 Theorem.......Page 59
9.5 Lemma.......Page 60
9.7 Theorem (Basic Examples). (a) lim 1......Page 61
9.8 Definiion.......Page 63
9.9 Theorem.......Page 65
9.10 Theorem.......Page 66
10.1 Definiion.......Page 69
10.2 Theorem.......Page 70
10.3 Discussion of Decimals.......Page 71
10.4 Theorem.......Page 72
10.6 Definiion.......Page 73
10.7 Theorem.......Page 74
10.8 Definiion.......Page 75
10.11 Theorem.......Page 76
11.1 Definiion.......Page 79
11.2 Theorem.......Page 81
11.4 Theorem.......Page 84
11.6 Definiion.......Page 85
11.7 Theorem.......Page 86
11.8 Theorem.......Page 87
11.9 Theorem.......Page 88
12.1 Theorem.......Page 91
12.2 Theorem.......Page 92
12.3 Corollary.......Page 94
§13 * Some Topological Concepts in Metric Spaces......Page 96
13.1 Definiion.......Page 97
13.3 Lemma.......Page 98
13.5 Bolzano-Weierstrass Theorem.......Page 99
13.7 Discussion.......Page 100
13.9 Proposition.......Page 101
13.11 Definiion.......Page 102
13.12 Heine-Borel Theorem.......Page 103
13.13 Proposition.......Page 104
14.2 Infinit Series.......Page 108
14.3 Definiion.......Page 110
14.6 Comparison Test.......Page 111
14.9 Root Test.......Page 112
14.10 Remark.......Page 114
§15 Alternating Series and Integral Tests......Page 118
15.2 Integral Tests.......Page 120
15.3 Alternating Series Theorem.......Page 121
§16 * Decimal Expansions of Real Numbers......Page 122
16.1 Long Division.......Page 123
16.2 Theorem.......Page 125
16.3 Theorem.......Page 126
16.4 Definiion.......Page 127
16.5 Theorem.......Page 128
§17 Continuous Functions......Page 136
17.2 Theorem.......Page 137
17.4 Theorem.......Page 141
17.5 Theorem.......Page 142
18.1 Theorem.......Page 146
18.2 Intermediate Value Theorem.......Page 147
18.3 Corollary.......Page 148
18.5 Theorem.......Page 150
18.6 Theorem.......Page 151
§19 Uniform Continuity......Page 152
19.1 Definiion.......Page 153
19.2 Theorem.......Page 156
19.3 Discussion.......Page 157
19.4 Theorem.......Page 159
19.5 Theorem.......Page 161
19.6 Theorem.......Page 163
20.2 Remarks. (a) From De.nition 17.1 we see that a function f is continuous at......Page 166
20.3 Definiion. (a) For a . R and a function f we write limx a f (x)=L provided......Page 167
20.4 Theorem.......Page 169
20.5 Theorem.......Page 171
20.7 Corollary.......Page 172
20.10 Theorem.......Page 173
20.11 Remark.......Page 175
21.1 Definiion.......Page 177
21.3 Theorem.......Page 180
21.4 Theorem.......Page 182
21.5 Corollary.......Page 183
21.6 Remark.......Page 184
21.7 Theorem.......Page 185
21.8 Baire Category Theorem.......Page 186
21.10 Discussion.......Page 187
21.11 Theorem.......Page 188
22.1 Definiion.......Page 191
22.5 Theorem.......Page 193
0) = r} and the k-dimensi{onal cube }......Page 194
22.6 Definiion.......Page 196
§23 Power Series......Page 200
23.1 Theorem.......Page 201
24.1 Definiion.......Page 206
24.2 Definiion.......Page 207
24.3 Theorem.......Page 209
24.4 Remark.......Page 210
§25 More on Uniform Convergence......Page 213
25.2 Theorem.......Page 214
25.4 Theorem.......Page 215
25.5 Theorem.......Page 217
25.7 Weierstrass M-test.......Page 218
26.1 Theorem.......Page 221
26.3 Lemma.......Page 222
26.5 Theorem.......Page 223
26.6 Abel’s Theorem.......Page 225
§27 * Weierstrass’s Approximation Theorem......Page 229
27.2 Lemma.......Page 230
27.4 Theorem.......Page 231
27.6 Corollary.......Page 233
28.1 Definiion.......Page 236
28.2 Theorem.......Page 238
28.3 Theorem.......Page 239
28.4 Theorem [Chain Rule].......Page 240
29.1 Theorem.......Page 245
29.3 Mean Value Theorem.......Page 246
29.4 Corollary.......Page 247
29.6 Definiion.......Page 248
29.8 Intermediate Value Theorem for Derivatives.......Page 249
29.9 Theorem.......Page 250
30.1 Generalized Mean Value Theorem.......Page 254
30.2 L’Hospital’s Rule.......Page 255
31.1 Discussion.......Page 262
31.3 Taylor’s Theorem.......Page 263
31.4 Corollary.......Page 264
31.5 Taylor’s Theorem.......Page 266
31.6 Corollary.......Page 267
31.7 Binomial Series Theorem.......Page 268
31.9 Secant Method.......Page 272
31.10 Lemma.......Page 275
31.11 Discussion.......Page 276
31.12 Theorem.......Page 277
31.13 Theorem.......Page 278
32.1 Definiion.......Page 282
32.3 Lemma.......Page 286
32.5 Theorem.......Page 287
32.7 Theorem.......Page 288
32.8 Definiion.......Page 289
32.9 Theorem.......Page 290
32.10 Corollary.......Page 291
32.11 Remark.......Page 292
33.1 Theorem.......Page 293
33.3 Theorem.......Page 294
33.5 Theorem.......Page 297
33.6 Theorem.......Page 298
33.8 Theorem.......Page 299
33.10 Discussion.......Page 300
33.12 Monotone Convergence Theorem.......Page 301
§34 Fundamental Theorem of Calculus......Page 304
34.1 Fundamental Theorem of Calculus I.......Page 305
34.2 Theorem [Integration by Parts].......Page 306
34.3 Fundamental Theorem of Calculus II.......Page 307
34.4 Theorem [Change of Variable].......Page 308
§35 * Riemann-Stieltjes Integrals......Page 311
35.1 Notation.......Page 312
35.2 Definiion.......Page 313
35.3 Lemma.......Page 316
35.7 Theorem.......Page 317
35.11 Theorem.......Page 318
35.12 Theorem.......Page 319
35.13 Theorem.......Page 323
35.14 Theorem.......Page 325
35.15 Lemma.......Page 326
35.16 Theorem.......Page 327
35.18 Proposition.......Page 328
35.19 Theorem [Integration by Parts].......Page 329
35.20 Theorem.......Page 331
35.24 Definiion.......Page 333
35.25 Theorem.......Page 334
35.26 Remarks.......Page 335
35.27 Proposition.......Page 336
35.28 Corollary.......Page 337
35.29 Theorem.......Page 338
35.30 Lemma.......Page 339
35.31 Corollary.......Page 340
35.32 Lemma.......Page 341
36.1 Definiion.......Page 344
36.2 Definiion.......Page 346
36.4 Theorem.......Page 347
37.1 Piecemeal Approach.......Page 352
37.2 Exponential Power Series Approach.......Page 354
37.3 Logarithmic Integral Approach.......Page 355
37.4 Theorem. (i) The function L is strictly increasing, continuous and differen- ......Page 356
37.6 Theorem.......Page 357
37.10 Theorem.......Page 358
37.12 Trigonometric Functions.......Page 359
§38 * Continuous Nowhere-Differentiable Functions ......Page 360
38.1 Van der Waerden’s Example.......Page 361
38.3 Theorem.......Page 363
38.4 Proposition.......Page 365
38.6 Lemma.......Page 366
38.8 Theorem.......Page 370
38.10 Lemma.......Page 372
38.11 Mark Lynch’s Construction.......Page 374
Appendix on Set Notation......Page 378
Selected Hints and Answers......Page 380
References......Page 410
Symbols Index......Page 416
Index......Page 418
