图书简介

To the student; To the instructor; 0. Paradoxes!; 1. Logical foundations; 2. Proof, and the natural numbers; 3. The integers, and the ordered field of rational numbers; 4. Induction, and well-ordering; 5. Sets; 6. Functions; 7. Inverse functions; 8. Some subsets of the real numbers; 9. The rational numbers are denumerable; 10. The uncountability of the real numbers; 11. The infinite; 12. The complete, ordered field of real numbers; 13. Further properties of real numbers; 14. Cluster points and related concepts; 15. The triangle inequality; 16. Infinite sequences; 17. Limits of sequences; 18. Divergence: the non-existence of a limit; 19. Four great theorems in real analysis; 20. Limit theorems for sequences; 21. Cauchy sequences and the Cauchy convergence criterion; 22. The limit superior and limit inferior of a sequence; 23. Limits of functions; 24. Continuity and discontinuity; 25. The sequential criterion for continuity; 26. Theorems about continuous functions; 27. Uniform continuity; 28. Infinite series of constants; 29. Series with positive terms; 30. Further tests for series with positive terms; 31. Series with negative terms; 32. Rearrangements of series; 33. Products of series; 34. The numbers e and γ; 35. The functions exp x and ln x; 36. The derivative; 37. Theorems for derivatives; 38. Other derivatives; 39. The mean value theorem; 40. Taylor’s theorem; 41. Infinite sequences of functions; 42. Infinite series of functions; 43. Power series; 44. Operations with power series; 45. Taylor series; 46. Taylor series, part II; 47. The Riemann integral; 48. The Riemann integral, part II; 49. The fundamental theorem of integral calculus; 50. Improper integrals; 51. The Cauchy–Schwartz and Minkowski inequalities; 52. Metric spaces; 53. Functions and limits in metric spaces; 54. Some topology of the real number line; 55. The Cantor ternary set; Appendix A. Farey sequences; Appendix B. Proving that; Appendix C. The ruler function is Riemann integrable; Appendix D. Continued fractions; Appendix E. L’Hospital’s Rule; Appendix F. Symbols, and the Greek alphabet; Bibliography; Solutions; Index.

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