Ⅰ-Sets and Functions 1
1．Set Theory 7
1-Membership,equality,empty set 7
2-The set defined by a relation.Intersections and unions 10
3-Whole numbers.Infinite sets 13
4-Ordered pairs,Cartesian products,sets of subsets 17
5-Functions,maps,correspondences 19
6-Injections,surjections,bijections 23
7-Equipotent sets.Countable sets 25
8-The different types of infinity 28
9-Ordinals and cardinals 31
2．The logic of logicians 39
Ⅱ-Convergence:Discrete variables 45
1．Convergent sequences and series 45
0-Introduction: what is a real number? 45
1-Algebraic operations and the order relation:axioms of R 53
2-Inequalities and intervals 56
3-Local or asymptotic properties 59
4-The concept of limit.Continuity and differentiability 63
5-Convergent sequences:definition and examples 67
6-The language of series 76
7-The marvels of the harmonic series 81
8-Algebraic operations on limits 95
2．Absolutely convergent series 98
9-Increasing sequences. Upper bound of a set of real numbers 98
10-The function logx. Roots of a positive number 103
11-What is an integral? 110
12-Series with positive terms 114
13-Alternating series 119
14-Classical absolutely convergent series 123
15-Unconditional convergence:general case 127
16-Comparison relations.Criteria of Cauchy and d'Alembert 132
17-Infinite limits 138
18-Unconditional convergence:associativity 139
3．First concepts of analytic functions 148
19-The Taylor series 148
20-The principle of analytic continuation 158
21-The function cot x and the series Σ1/n2k 162
22-Multiplication of series.Composition of analytic func-tions.Formal series 167
23-The elliptic functions of Weierstrass 178
Ⅲ-Convergence:Continuous variables 187
1．The intermediate value theorem 187
1-Limit values of a function.Open and closed sets 187
2-Continuous functions 192
3-Right and left limits of a monotone function 197
4-The intermediate value theorem 200
2．Uniform convergence 205
5-Limits of continuous functions 205
6-A slip up of Cauchy's 211
7-The uniform metric 216
8-Series of continuous functions.Normal convergence 220
3．Bolzano-Weierstrass and Cauchy's criterion 225
9-Nested intervals,Bolzano-Weierstrass,compact sets 225
10-Cauchy's general convergence criterion 228
11-Cauchy's criterion for series:examples 234
12-Limits of limits 239
13-Passing to the limit in a series of functions 241
4．Differentiable functions 244
14-Derivatives of a function 244
15-Rules for calculating derivatives 252
16-The mean value theorem 260
17-Sequences and series of differentiable functions 265
18-Extensions to unconditional convergence 270
5．Differentiable functions of several variables 273
19-Partial derivatives and differentials 273
20-Differentiability of functions of class C1 276
21-Diferentiation of composite functions 279
22-Limits of differentiable functions 284
23-Interchanging the order of differentiation 287
24-Implicit functions 290
Appendix to Chapter Ⅲ 303
1-Cartesian spaces and general metric spaces 303
2-Open and closed sets 306
3-Limits and Cauchy's criterion in a metric space;complete spaces 308
4-Continuous functions 311
5-Absolutely convergent series in a Banach space 313
6-Continuous linear maps 316
7-Compact spaces 320
8-Topological spaces 322
Ⅳ-Powers,Exponentials,Logarithms,Trigonometric Functions 325
1．Direct construction 325
1-Rational exponents 325
2-Definition of real powers 327
3-The calculus of real exponents 330
4-Logarithms to base a.Power functions 332
5-Asymptotic behaviour 333
6-Characterisations of the exponential,power and logarithmic functions 336
7-Derivatives of the exponential functions:direct method 339
8-Derivatives of exponential functions,powers and logarithms 342
2．Series expansions 345
9-The number e.Napierian logarithms 345
10-Exponential and logarithmic series:direct method 346
11-Newton's binomial series 351
12-The power series for the logarithm 359
13-The exponential function as a limit 368
14-Imaginary exponentials and trigonometric functions 372
15-Euler's relation chez Euler 383
16-Hyperbolic functions 388
3．Infinite products 394
17-Absolutely convergent infinite products 394
18-The infinite product for the sine function 397
19-Expansion of an infinite product in series 403
20-Strange identities 407
4．The topology of the functions Arg(z)and Log z 414
Index 425