مسائل و راه‌حل‌ها برای تحلیل واقعی دورهٔ کارشناسی I

The aim of Problems and Solutions for Undergraduate Real Analysis I , as the name reveals, is to assist undergraduate students or first-year students who study mathematics in learning their first rigorous real analysis course. The wide variety of problems, which are of varying difficulty, include the following topics: Elementary Set Algebra The Real Number System Countable and Uncountable Sets Elementary Topology on Metric Spaces Sequences in Metric Spaces Series of Numbers Limits and Continuity of Functions Differentiation the Riemann-Stieltjes Integral Furthermore, the main features of this book are listed as follows: The book contains 230 problems, which cover the topics mentioned above, with detailed and complete solutions. As a matter of fact, my solutions show every detail, every step and every theorem that I applied. Each chapter starts with a brief and concise note of introducing the notations, terminologies, basic mathematical concepts or important/famous/frequently used theorems (without proofs) relevant to the topic. Three levels of difficulty have been assigned to problems so that you can sharpen your mathematics step-by-step. Different colors are used frequently in order to highlight or explain problems, examples, remarks, main points/formulas involved, or show the steps of manipulation in some complicated proofs. (ebook only) An appendix about mathematical logic is included. It tells students what concepts of logic (e.g. techniques of proofs) are necessary in advanced mathematics. Preface 5 List of Figures 7 List of Tables 9 Contents 11 Chapter 1. Elementary Set Algebra 15 1.1 Fundamental Concepts 15 1.2 Sets, Functions and Relations 18 Chapter 2. The Real Number System 23 2.1 Fundamental Concepts 23 2.2 Rational and Irrational Numbers 24 2.3 Absolute Values 26 2.4 The Completeness Axiom 27 Chapter 3. Countable and Uncountable Sets 33 3.1 Fundamental Concepts 33 3.2 Problems on Countable and Uncountable Sets 34 Chapter 4. Elementary Topology on Metric Spaces 41 4.1 Fundamental Concepts 41 4.2 Open Sets and Closed Sets 45 4.3 Compact Sets 52 4.4 The Heine-Borel Theorem 58 4.5 Connected Sets 59 Chapter 5. Sequences in Metric Spaces 63 5.1 Fundamental Concepts 63 5.2 Convergence of Sequences 67 5.3 Upper and Lower Limits 73 5.4 Cauchy Sequences and Complete Metric Spaces 79 5.5 Recurrence Relations 84 Chapter 6. Series of Numbers 89 6.1 Fundamental Concepts 89 6.2 Convergence of Series of Nonnegative Terms 93 6.3 Alternating Series and Absolute Convergence 101 6.4 The Series Σ_{n=1}^∞ a_n b_n and Multiplication of Series 104 6.5 Power Series 108 Chapter 7. Limits and Continuity of Functions 111 7.1 Fundamental Concepts 111 7.2 Limits of Functions 117 7.3 Continuity and Uniform Continuity of Functions 122 7.4 The Extreme Value Theorem and the Intermediate Value Theorem 130 7.5 Discontinuity of Functions 134 7.6 Monotonic Functions 136 Chapter 8. Differentiation 141 8.1 Fundamental Concepts 141 8.2 Properties of Derivatives 146 8.3 The Mean Value Theorem for Derivatives 152 8.4 L’Hôspital’s Rule 160 8.5 Higher Order Derivatives and Taylor’s Theorem 163 Chapter 9. The Riemann-Stieltjes Integral 171 9.1 Fundamental Concepts 171 9.2 Integrability of Real Functions 176 9.3 Applications of Integration Theorems 184 9.4 The Mean Value Theorems for Integrals 196 Chapter 10. Sequences and Series of Functions 201 10.1 Fundamental Concepts 201 10.2 Uniform Convergence for Sequences of Functions 206 10.3 Uniform Convergence for Series of Functions 215 10.4 Equicontinuous Families of Functions 223 10.5 Approximation by Polynomials 228 Chapter 11. Improper Integrals 233 11.1 Fundamental Concepts 233 11.2 Evaluations of Improper Integrals 237 11.3 Convergence of Improper Integrals 242 11.4 Miscellaneous Problems on Improper Integrals 250 Chapter 12. Lebesgue Measure 257 12.1 Fundamental Concepts 257 12.2 Lebesgue Outer Measure 261 12.3 Lebesgue Measurable Sets 265 12.4 Necessary and Sufficient Conditions for Measurable Sets 278 Chapter 13. Lebesgue Measurable Functions 283 13.1 Fundamental Concepts 283 13.2 Lebesgue Measurable Functions 285 13.3 Applications of Littlewood’s Three Principles 296 Chapter 14. Lebesgue Integration 305 14.1 Fundamental Concepts 305 14.2 Properties of Integrable Functions 309 14.3 Applications of Fatou’s Lemma 323 Chapter 15. Differential Calculus of Functions of Several Variables 341 15.1 Fundamental Concepts 341 15.2 Differentiation of Functions of Several Variables 347 15.3 The Mean Value Theorem for Differentiable Functions 355 15.4 The Inverse Function Theorem and the Implicit Function Theorem 357 15.5 Higher Order Derivatives 361 Chapter 16. Integral Calculus of Functions of Several Variables 367 16.1 Fundamental Concepts 367 16.2 Jordan Measurable Sets 374 16.3 Integration on R^n 378 16.4 Applications of the Mean Value Theorem 386 16.5 Applications of the Change of Variables Theorem 389 Appendix 397 A. Language of Mathematics 397 A.1 Fundamental Concepts 397 A.2 Statements and Logical Connectives 398 A.3 Quantifiers and their Basic Properties 400 A.4 Necessity and Sufficiency 401 A.5 Techniques of Proofs 402 Index 405 Bibliography 411 [1]-[16] 411 [17]-[34] 412