Contents ∗
Preface ix
About the Author xi
Chapter 1. Set Theory 1
1–3. Sets and Operations on Sets. Quantifiers . . . . . . . . . . . . . . . . . . . . . . . . . . .1
Problems in Set Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
4–7. Relations. Mappings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8
Problems on Relations and Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . .14
8. Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
9. Some Theorems on Countable Sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .18
Problems on Countable and Uncountable Sets . . . . . . . . . . . . . . . . . . 21
Chapter 2. Real Numbers. Fields 23
1–4. Axioms and Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
5–6. Natural Numbers. Induction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .27
Problems on Natural Numbers and Induction . . . . . . . . . . . . . . . . . . . 32
7. Integers and Rationals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .34
8–9. Upper and Lower Bounds. Completeness. . . . . . . . . . . . . . . . . . . . . . . . . .36
Problems on Upper and Lower Bounds . . . . . . . . . . . . . . . . . . . . . . . . . 40
10. Some Consequences of the Completeness Axiom. . . . . . . . . . . . . . . . . . .43
11–12. Powers With Arbitrary Real Exponents. Irrationals . . . . . . . . . . . . . . . 46
Problems on Roots, Powers, and Irrationals.. . . . . . . . . . . . . . . . . . . .50
13. The Infinities. Upper and Lower Limits of Sequences . . . . . . . . . . . . . .53
Problems on Upper and Lower Limits of Sequences in E∗ . . . . . . . 60
Chapter 3. Vector Spaces. Metric Spaces 63
1–3. The Euclidean n-space, E n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
Problems on Vectors in E n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4–6. Lines and Planes in E n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
Problems on Lines and Planes in E n . . . . . . . . . . . . . . . . . . . . . . . . . . . .75
∗ “Starred” sections may be omitted by beginners.vi Contents
7. Intervals in E n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
Problems on Intervals in E n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .79
8. Complex Numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .80
Problems on Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
∗9. Vector Spaces. The Space C n . Euclidean Spaces . . . . . . . . . . . . . . . . . . 85
Problems on Linear Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
∗10. Normed Linear Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .90
Problems on Normed Linear Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . .93
11. Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
Problems on Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
12. Open and Closed Sets. Neighborhoods . . . . . . . . . . . . . . . . . . . . . . . . . . .101
Problems on Neighborhoods, Open and Closed Sets. . . . . . . . . . . .106
13. Bounded Sets. Diameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .108
Problems on Boundedness and Diameters. . . . . . . . . . . . . . . . . . . . . .112
14. Cluster Points. Convergent Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
Problems on Cluster Points and Convergence . . . . . . . . . . . . . . . . . . 118
15. Operations on Convergent Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
Problems on Limits of Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
16. More on Cluster Points and Closed Sets. Density . . . . . . . . . . . . . . . . 135
Problems on Cluster Points, Closed Sets, and Density. . . . . . . . . .139
17. Cauchy Sequences. Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
Problems on Cauchy Sequences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .144
Chapter 4. Function Limits and Continuity 149
1. Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
Problems on Limits and Continuity. . . . . . . . . . . . . . . . . . . . . . . . . . . .157
2. Some General Theorems on Limits and Continuity . . . . . . . . . . . . . . . 161
More Problems on Limits and Continuity . . . . . . . . . . . . . . . . . . . . . .166
3. Operations on Limits. Rational Functions . . . . . . . . . . . . . . . . . . . . . . . 170
Problems on Continuity of Vector-Valued Functions. . . . . . . . . . . .174
4. Infinite Limits. Operations in E∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
Problems on Limits and Operations in E∗ . . . . . . . . . . . . . . . . . . . . . 180
5. Monotone Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
Problems on Monotone Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
6. Compact Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
Problems on Compact Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
∗7. More on Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
Contents vii
8. Continuity on Compact Sets. Uniform Continuity . . . . . . . . . . . . . . . .194
Problems on Uniform Continuity; Continuity on Compact Sets .200
9. The Intermediate Value Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
Problems on the Darboux Property and Related Topics . . . . . . . . 209
10. Arcs and Curves. Connected Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .211
Problems on Arcs, Curves, and Connected Sets . . . . . . . . . . . . . . . . 215
∗11. Product Spaces. Double and Iterated Limits . . . . . . . . . . . . . . . . . . . . . 218
∗Problems on Double Limits and Product Spaces . . . . . . . . . . . . . . 224
12. Sequences and Series of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
Problems on Sequences and Series of Functions . . . . . . . . . . . . . . . . 233
13. Absolutely Convergent Series. Power Series . . . . . . . . . . . . . . . . . . . . . . 237
More Problems on Series of Functions . . . . . . . . . . . . . . . . . . . . . . . . . 245
Chapter 5. Differentiation and Antidifferentiation 251
1. Derivatives of Functions of One Real Variable. . . . . . . . . . . . . . . . . . . .251
Problems on Derived Functions in One Variable . . . . . . . . . . . . . . . 257
2. Derivatives of Extended-Real Functions . . . . . . . . . . . . . . . . . . . . . . . . . .259
Problems on Derivatives of Extended-Real Functions . . . . . . . . . . 265
3. L’Hˆopital’s Rule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .266
Problems on L’Hˆopital’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
4. Complex and Vector-Valued Functions on E 1 . . . . . . . . . . . . . . . . . . . . 271
Problems on Complex and Vector-Valued Functions on E 1 . . . . . 275
5. Antiderivatives (Primitives, Integrals). . . . . . . . . . . . . . . . . . . . . . . . . . . .278
Problems on Antiderivatives. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .285
6. Differentials. Taylor’s Theorem and Taylor’s Series. . . . . . . . . . . . . . .288
Problems on Taylor’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296
7. The Total Variation (Length) of a Function f : E 1 → E . . . . . . . . . . 300
Problems on Total Variation and Graph Length . . . . . . . . . . . . . . . 306
8. Rectifiable Arcs. Absolute Continuity. . . . . . . . . . . . . . . . . . . . . . . . . . . .308
Problems on Absolute Continuity and Rectifiable Arcs . . . . . . . . . 314
9. Convergence Theorems in Differentiation and Integration . . . . . . . . 314
Problems on Convergence in Differentiation and Integration. . . .321
10. Sufficient Condition of Integrability. Regulated Functions . . . . . . . . 322
Problems on Regulated Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329
11. Integral Definitions of Some Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 331
Problems on Exponential and Trigonometric Functions . . . . . . . . 338
Index 341
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