Elements of Real Analysis (eBook)

PrefacePart I. Fundamental Concepts 1. Sets and Functions 1. Sets 2. The Algebra of Sets 3. Functions 4. Countability 2. The Rational Numbers 5. Alebraic Properties 6. Decimal ExpansionsPart II. The Real Line 3. The Real Number System 7. Cauchy Sequences and Their Equivalence Classes 8. The Real Number System 9. Completeness Properties of R 10. The Extended Real Line 4. Sequences and Series of Number 11. Sequences: Basic Limit Theorems 12. Upper and Lower Limits 13. Basic Properties of Series 14. Series with Nonnegative Terms 15. Alternating Series 16. Absolute Convergence 5. The Structure of Point Sets 17. Basic Notions 18. Closed Sets 19. Open Sets 20. Perfect Sets 21. Distance between Point Sets 22. Connected Sets 3. Functions of a Real Variable 23. Continuity; Limits of Functions 24. Continuous Functions 25. The Nature of Discontinuities 26. Monotonic Functions 27. Uniform Continuity 7. Differentiability 28 The Derivative at a Point 29. A Continuous Nowhere Differentiable Function 30. Properties of the Derivative 31. Taylor's Theorem 8. Spaces of Continuous Functions 32. The Problems of Separability and Convergence 33. Uniform Convergence 34. Power Series 35. The Approximation of Functions 36. Equicontinuity 37. Summary 9. Measure and Integration 38. Measurable Sets 39. Properties of Measurable Sets 40. Measurable Functions 41. The Lebesgue Integral of Simple Functions 42. The Lebesgue Integral 43. Theorems on Limits under the Integral Sign 44. The Riemann Integral 10. Fourier Series 45. Basic Facts 46. The Space £ superscript 2 47. The Question of Convergence Bibliography; Glossary of Symbols; Index