Mathematical Analysis Vol 1, (2011)

* “Starred” sections may be omitted by beginners.

Chapter 1. Set Theory

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, En 63
Problems on Vectors in En 69
4–6. Lines and Planes in En 71
Problems on Lines and Planes in En 75
7. Intervals in En 76
Problems on Intervals in En 79
8. Complex Numbers 80
Problems on Complex Numbers 83
*9. Vector Spaces. The Space Cn. 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
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 E1 271
Problems on Complex and Vector-Valued Functions on E1 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 : E1 ? 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