Preface to first edition xii
Preface to second edition xxi
Glossary xxii
PART I NUMBERS
1 Mathematical induction 3 Mathematical induction (qns 1—8) 3
Historical note 6 Answers and comments 8
2 Inequalities 10 Positive numbers and their properties (qns 1—29) 10
Summary — properties of order 14
Arithmetic mean and geometric mean (qns 30—39) 14
Completing the square (qns 40—42) 16
The sequence (1 Y 1/n)v (qns 43—49) 17
nth roots (qns 50, 51) 18
Summary — results on inequalities 18
Absolute value (qns52—64) 19
Summary — results on absolute value 21
Historical note 21
Answers and comments 22
3 Sequences: a first bite at infinity 28
Monotonic sequences (qn 4) 29
Bounded sequences (qns 5—7) 30
Subsequences (qns 8—16) 31
Sequences tending to infinity (qns 17, 18) 35
Archimedean order and the integer function (qns 19—23) 36
Summary — the language of sequences 37
Null sequences (qns 24—47) 38
Summary — null sequences 44
Convergent sequences and their limits (qns 48—60) 45
Boundedness of convergent sequences (qns 61—63) 49
Quotients of convergent sequences (qns 64—69) 49
d’Alembert’s ratio test (qns 70—74) 50
Convergent sequences in closed intervals (qns 75—80) 52
Intuition and convergence (qns 81—83) 53
Summary — convergent sequences 55
Historical note 57
Answers and comments 59
4 Completeness: what the rational numbers lack 68
The Fundamental Theorem of Arithmetic (qns 1—3) 68
Dense sets of rational numbers on the number line (qns 4—10) 69
Infinite decimals (qns11—17) 70
Irrational numbers (qns 18—21) 72
Infinity: countability (qns 22—30) 73 Summary 75
The completeness principle: infinite decimals are convergent (qns 31, 32) 76
Bounded monotonic sequences (qns 33—36) 78
nth roots of positive numbers, n a positive integar (qns37—41) 79
Nested closed intervals (qn 42) 80
Convergent subsequences of bonded sequences (qns 43—47) 81
Cluster points (the Bolzano—Weierstrass theorem) (qns 48—54) 81
Cauchy sequences (qns 55—58) 83
Least upper bounds (sup) and greatest lower bounds (inf) (qns 59—82) 85
Upper bounds and greatest terms (qns 59—61) 85
Least upper bound (sup) (qns 62—66) 86
Lower bounds and least members (qns 67—70) 87
Greatest lower bound (inf) (qns 71—78) 88
sup, inf and completeness (qns 79—82) 89
lim sup and lim inf (qns 83, 84) 90
Summary — completeness 91
Historical note 92
Answers and comments 95
5 Series: infinite sums 104
Sequences of partial sums (qns 1—10) 104
The null sequence test (qns 11—13) 107
Simple consequences of convergence (qns 14—22) 107
Summary — convergence of series 108
Series of positive terms 109
First comparison test (qns 23—29) 109
The harmonic series (qn 30) 110
The convergence of Σ1/n^α (qns 31, 32) 110
Cauchy’s nth root test (qns 33—39) 111
d’Alembert’s ratio test (qns 40—50) 112
Second comparison test (qns 51—55) 113
Integral test (qns 56—61) 114
Summary — series of positive terms 116
Series with positive and negative terms 117
Alternating series test (qns 62—65) 117
Absolute convergence (qns 66 —70) 118
Conditional convergence (qn 71) 119
Rearrangements (qns 72—77) 120
Summary — series of positive and negative terms 122
Power series 123
Application of d’Alembert’s ratio test and Cauchy’s nth
root test for absolute convergence (qns 78—90) 123
Radius of convergence (qns 91—101) 123
Cauchy—Hadamard formula (qns 102—107) 125
The Cauchy product(qns108—113) 126
Summary — power series and the Cauchy product 128
Historical note 129
Answers and comments 131
PART II FUNCTIONS
6 Functionsandcontinuity:neighbourhoods,limits of functions 143
Functions (qn 1) 143
The domain of a function 143
The range and co-domain of a function(qns2—4) 144
Bijections and inverse functions (qns 5—6) 145
Summary — functions 145
Continuity (qns 7—11)
Definition of continuity by sequences (qns 12—18) 147
Examples of discontinuity (qns 19, 20) 148
Sums and products of continuous functions (qns 21—31) 149
Continuity in less familiar settings (qns 32—35) 150
A squeeze rule (qns 36, 37) 151
Continuity of composite functions and quotients of
continuous functions (qns 38—55) 151
Summary — continuity by sequences 154
Neighbourhoods (qns 56—63) 154
Definition of continuity by neighbourhoods (qns 64—72) 156
One-sided limits 159
Definition of one-sided limits by sequences (qns 73—82) 159
Definition of one-sided limits by neighbourhoods (qns 83—85) 162
Two-sided limits 163 Definition of continuity by limits (qns 86—92) 163
Theorems on limits (qns 93—99) 164
Limits as xYE and when f(x)YE (qns 100, 101) 166
Summary — continuity by neighbourhoods and limits 167
Historical note 168 Answers 171
7 Continuity and completeness: functions on intervals 182
Monotonic functions: one-sided limits (qns 1—7) 182
Intervals (qns 8—11) 183
Intermediate Value Theorem (qns 12—21) 185
Inverses of continuous functions (qns 22—28) 186
Continuous functions on a closed interval (qns 29—36) 188
Uniform continuity (qn 37—45) 190
Extension of functions on Q to functions on R (qns 46—48) 192
Summary 194
Historical note 195
Answers 197
8 Derivatives: tangents 203
Definition of derivative (qns 1—9) 203
Sums of functions (qns 10, 11) 205
The product rule(qns13—16) 205
The quotient rule(qn17) 206
The chain rule (qn18) 206
Differentiability and continuity (qns 12, 19—25) 207
Derived functions (qns 26—34) 211
Second derivatives (qns 35—38) 212
Inverse functions (qns 39—45) 213
Derivatives at end points (qn 46) 215
Summary 215
Historical note 216 Answers 219
9 Differentiation and completeness: Mean Value Theorems,Taylor’s Theorem 224
Rolle’s Theorem (qns 1—11) 224
An intermediate value theorem for derivatives (qn 12) 226
The Mean Value Theorem (qns 13—24) 226
Cauchy’s Mean Value Theorem (qn 25) 230
de l’Hoˆpital’s rule(qns26—30) 230
Summary — Rolle’s Theorem and Mean Value Theorem 232
The Second and Third Mean Value Theorems (qns 31—34) 234
Taylor’s Theorem or nth Mean Value Theorem (qns 35, 36) 235
Maclaurin’s Theorem (qns 37—46) 236
Summary — Taylor’s Theorem 239
Historical note 240
Answers 243
10 Integration: the Fundamental Theorem of Calculus 251
Areas with curved boundaries (qns 1—6) 251
Monotonic functions (qns 7—9) 254
The definite integral (qns 10—11) 255
Step functions (qns 12—15) 256
Lower integral and upper integral (qns 16—22) 258
The Riemann integral (qns 23—25) 260
Summary — definiton of the Riemann integral 261
Theorems on integrability (qns 26—36) 262
Integration and continuity (qns 37—44) 265
Mean Value Theorem for integrals (qn 45) 267
Integration on subintervals (qns 46—48) 267
Summary — properties of the Riemann integral 267
Indefinite integrals (qns 49—53) 268
The Fundamental Theorem of Calculus (qns 54—56) 270
Integration by parts (qns 57—59) 270
Integration by substitution (qn 60) 271
Improper integrals (qns 61—68) 271
Summary — the Fundamental Theorem of Calcalus 273
Historical note 273
Answers 276
11 Indices and circle functions 286
Exponential and logarithmic functions 286
Positive integers as indices (qns 1—3) 286
Positive rationals as indices(qns4—7) 287
Rational numbers as indices (qns 8 —17) 287
Real numbers as indices (qns 18—24) 289
Natural logarithms (qns 25—31) 290
Exponential and logarithmic limits (qns 32—38) 291
Summary — exponential and logarithmic functions 292
Circular or trigonometric functions 293
Length of a line segment (qns 39—42) 294
Arc length (qns 43—48) 295
Arc cosine(qns49,50) 296
Cosine and sine(qns51—58) 297
Tangent (qns 59—62) 298
Summary — circular or trigonometric functions Historical note 298
Answers 302
12 Sequences of functions 309
Pointwise limit functions (qns1—14) 310
Uniform convergence (qns 15—19) 312
Uniform convergence and continuity (qns 20—23) 313
Uniform convergence and integration (qns 24—31) 315
Summary — uniform convergence, continuity and integration 317
Uniform convergence and differentiation (qns 32—34) 318
Uniform convergence of power series (qns 35—44) 319
The Binomial Theorem for any real index (qn 45) 322
The blancmange function (qn46) 322
Summary — differentiation and the M-test 326
Historical note 326
Answers 328
Appendix 1 Properties of the real numbers 337
Appendix 2 Geometry and intuition 340
Appendix 3 Questions for student investigation and discussion 342
Bibliography 346
Index 351