Numbers and Functions: Steps into Analysis, Second Edition

Cover......Page 1
Half-title......Page 3
Series-title......Page 4
Title......Page 5
Copyright......Page 6
Contents......Page 7
Preface to first edition......Page 14
Preface to second edition......Page 23
Glossary......Page 24
PART I Numbers......Page 27
1 Mathematical induction......Page 29
Historical note......Page 32
Answers and comments......Page 34
Positive numbers and their properties......Page 36
Arithmetic mean and geometric mean......Page 40
Completing the square......Page 42
The sequence (1+ 1/n)......Page 43
Summary – results on inequalities......Page 44
Absolute value......Page 45
Historical note......Page 47
Answers and comments......Page 48
3 Sequences A first bite at infinity......Page 54
Monotonic sequences......Page 55
Bounded sequences......Page 56
Subsequences......Page 57
Sequences tending to infinity......Page 61
Archimedean order and the integer function......Page 62
Summary – the language of sequences......Page 63
Null sequences......Page 64
Summary – null sequences......Page 70
Convergent sequences and their limits......Page 71
Quotients of convergent sequences......Page 75
d\'Alembert\'s ratio test......Page 76
Convergent sequences in closed intervals......Page 78
Intuition and convergence......Page 79
Summary – convergent sequences......Page 82
Historical note......Page 83
Answers and comments......Page 85
The Fundamental Theorem of Arithmetic......Page 94
Dense sets of rational numbers on the number line......Page 95
Infinite decimals......Page 96
Irrational numbers......Page 98
Infinity: countability......Page 99
Decimals and irrationals......Page 101
The completeness principle: infinite decimals are convergent......Page 102
Every infinite decimal sequence is convergent.......Page 103
Bounded monotonic sequences......Page 104
nth roots of positive real numbers, n a positive integer......Page 105
Nested closed intervals......Page 106
Cluster points (the Bolzano–Weierstrass theorem)......Page 107
Cauchy sequences......Page 109
Upper bounds and greatest terms......Page 111
Least upper bound (sup)......Page 112
Lower bounds and least members......Page 113
Greatest lower bound (inf)......Page 114
sup, inf and completeness......Page 115
lim sup and lim inf......Page 116
Summary – completeness......Page 117
Historical note......Page 118
Answers and comments......Page 121
Sequences of partial sums......Page 130
Simple consequences of convergence......Page 133
Summary – convergence of series......Page 134
First comparison test......Page 135
The convergence of Sigma 1/n......Page 136
Cauchy’s nth root test......Page 137
d\'Alembert’s ratio test......Page 138
Second comparison test......Page 139
Integral test......Page 140
Summary – series of positive terms......Page 142
Alternating series test......Page 143
Absolute convergence......Page 144
Conditional convergence......Page 145
Rearrangements......Page 146
Summary – series of positive and negative terms......Page 148
Radius of convergence......Page 149
Cauchy–Hadamard formula......Page 151
The Cauchy product......Page 152
Summary – power series and the Cauchy product......Page 154
Historical note......Page 155
Answers and comments......Page 157
PART II Functions......Page 167
The domain of a function......Page 169
The range and co-domain of a function......Page 170
Summary – functions......Page 171
Continuity......Page 172
Definition of continuity by sequences......Page 173
Examples of discontinuity......Page 174
Sums and products of continuous functions......Page 175
Continuity in less familiar settings......Page 176
Continuity of composite functions and quotients of continuous functions......Page 177
Neighbourhoods......Page 180
Definition of one-sided limits by sequences......Page 185
Definition of one-sided limits by neighbourhoods......Page 188
Definition of continuity by limits......Page 189
Theorems on limits......Page 190
Limits as …......Page 192
Summary – continuity by neighbourhoods and limits......Page 193
Historical note......Page 194
Answers......Page 197
Monotonic functions: one-sided limits......Page 208
Intervals......Page 209
Intermediate Value Theorem......Page 211
Inverses of continuous functions......Page 213
Continuous functions on a closed interval......Page 214
Uniform continuity......Page 216
Extension of functions on Q to functions on R......Page 218
Summary......Page 220
Historical note......Page 221
Answers......Page 223
8 Derivatives Tangents......Page 229
The product rule......Page 231
The chain rule......Page 232
Differentiability and continuity......Page 233
Derived functions......Page 237
Second derivatives......Page 238
Inverse functions......Page 239
Summary......Page 241
Historical note......Page 242
Answers......Page 245
Rolle’s Theorem......Page 250
The Mean Value Theorem......Page 252
de l\'Hôpital’s rule......Page 256
Summary – Rolle’s Theorem and Mean Value Theorem......Page 258
The Second and Third Mean Value Theorems......Page 260
Maclaurin’s Theorem......Page 261
Taylor’s Theorem with Cauchy’s form of the remainder......Page 265
Historical note......Page 266
Answers......Page 269
Areas with curved boundaries......Page 277
Monotonic functions......Page 280
The definite integral......Page 281
Step functions......Page 282
Lower integral and upper integral......Page 284
The Riemann integral......Page 286
Step functions......Page 287
Theorems on integrability......Page 288
Integration and continuity......Page 291
Summary – Properties of the Riemann integral......Page 293
Indefinite integrals......Page 294
Integration by parts......Page 296
Improper integrals......Page 297
Historical note......Page 299
Answers......Page 302
Positive integers as indices......Page 312
Rational numbers as indices......Page 313
Real numbers as indices......Page 315
Natural logarithms......Page 316
Exponential and logarithmic limits......Page 317
Summary – Exponential and logarithmic functions......Page 318
Circular or trigonometric functions......Page 319
Length of a line segment......Page 320
Arc length......Page 321
Arc cosine......Page 322
Cosine and sine......Page 323
Summary – Circular or trigonometric functions......Page 324
Historical note......Page 325
Answers......Page 328
12 Sequences of functions......Page 335
Pointwise limit functions......Page 336
Uniform convergence......Page 338
Uniform convergence and continuity......Page 339
Uniform convergence and integration......Page 341
Summary – Uniform convergence, continuity and integration......Page 343
Uniform convergence and differentiation......Page 344
Uniform convergence of power series......Page 345
The blancmange function......Page 348
Historical note......Page 352
Answers......Page 354
Appendices......Page 361
Order properties of a field of numbers – chapter 2 onwards......Page 363
Principle of completeness – chapter 4 onwards......Page 364
Appendix 2 Geometry and intuition......Page 366
Multiple-choice questions for student discussion......Page 368
Problems for corporate or individual investigation......Page 369
Bibliography......Page 372
Index......Page 377