Preface ........................................................ xi
To the Student ................................................ xxi
To the Instructor ........................................... xxvii
1 Construction of the Real Numbers ............................. 1
1.1 Introduction ............................................ 1
1.2 Entry 1: Axioms for the Natural Numbers ................. 2
1.3 Constructing the Integers .............................. 11
1.4 Entry 2: Axioms for the Integers ....................... 19
1.5 Constructing the Rational Numbers ...................... 27
1.6 DedekindCuts ........................................... 33
1.7 Constructing the Real Numbers .......................... 41
1.8 Historical Remarks ..................................... 51
2 Properties of the Real Numbers .............................. 61
2.1 Introduction ........................................... 61
2.2 Entry 3: Axioms for the Real Numbers ................... 62
2.3 Algebraic Properties of the Real Numbers ............... 65
2.4 Finding the Natural Numbers, the Integers and the
Rational Numbers in the Real Numbers ................... 75
2.5 Induction and Recursion in Practice .................... 83
2.6 The Least Upper Bound Property and Its Consequences .... 96
2.7 Uniqueness of the Real Numbers ........................ 107
2.8 Decimal Expansion of Real Numbers ..................... 113
2.9 Historical Remarks .................................... 128
3 Limits and Continuity ...................................... 129
3.1 Introduction .......................................... 129
3.2 Limits of Functions ................................... 129
3.3 Continuity ............................................ 146
3.4 Uniform Continuity .................................... 156
3.5 Two Important Theorems ................................ 163
3.6 Historical Remarks .................................... 171
4 Differentiation ............................................ 181
4.1 Introduction .......................................... 181
4.2 The Derivative ........................................ 181
4.3 Computing Derivatives ................................. 192
4.4 The Mean Value Theorem ................................ 198
4.5 Increasing and Decreasing Functions, Part I: Local
and Global Extrema .................................... 207
4.6 Increasing and Decreasing Functions, Part II:
Further Topics ........................................ 215
4.7 Historical Remarks .................................... 225
5 Integration ................................................ 231
5.1 Introduction .......................................... 231
5.2 The Riemann Integral .................................. 231
5.3 Elementary Properties of the Riemann Integral ......... 242
5.4 Upper Sums and Lower Sums ............................. 247
5.5 Further Properties of the Riemann Integral ............ 258
5.6 Fundamental Theorem of Calculus ....................... 267
5.7 Computing Antiderivatives ............................. 277
5.8 Lebesgue's Theorem .................................... 283
5.9 Area and Arc Length ................................... 293
5.10 Historical Remarks .................................... 312
6 Limits to Infinity ......................................... 321
6.1 Introduction .......................................... 321
6.2 Limits to Infinity .................................... 322
6.3 Computing Limits to Infinity .......................... 331
6.4 Improper Integrals .................................... 341
6.5 Historical Remarks .................................... 354
7 Transcendental Functions ................................... 357
7.1 Introduction .......................................... 357
7.2 Logarithmic and Exponential Functions ................. 358
7.3 Trigonometric Functions ............................... 369
7.4 More about ............................................ 379
7.5 Historical Remarks .................................... 391
8 Sequences .................................................. 399
8.1 Introduction .......................................... 399
8.2 Sequences ............................................. 399
8.3 Three Important Theorems .............................. 412
8.4 Applications of Sequences ............................. 423
8.5 Historical Remarks .................................... 439
9 Series ..................................................... 443
9.1 Introduction .......................................... 443
9.2 Series ................................................ 443
9.3 Convergence Tests ..................................... 451
9.4 Absolute Convergence and Conditional Convergence ...... 459
9.5 Power Series as Functions ............................. 473
9.6 Historical Remarks .................................... 482
10 Sequences and Series of Functions .......................... 489
10.1 Introduction .......................................... 489
10.2 Sequences of Functions ................................ 489
10.3 Series of Functions ................................... 502
10.4 Functions as Power Series ............................. 509
10.5 A Continuous but Nowhere Differentiable Function ...... 527
10.6 Historical Remarks .................................... 534
Bibliography .................................................. 539
Index ......................................................... 545
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