A large number of examples is included, with hints for the solution of many of them. These will be of particular value to students working on their own.
A large number of examples is included, with hints for the solution of many of them. These will be of particular value to students working on their own.
Preface Notation and Conventions Preliminaries 1. Enumerability and sequences 2. Bounds for sets of numbers 3. Bounds for functions and sequences 4. Limits of the sequences 5. Monotonic sequences two important examples irrational powers of positive real numbers 6. Upper and lower limits of real sequences: the general principle of convergence 7. Convergence of series absolute convergence 8. Conditional convergence 9. Rearrangement and multiplication of absolutely convergent series 10. Double series 11. Power series 12. Point set theory 13. The Bolzano-Weierstrass, Cantor and Heine-Borel theorems 14. Functions defined over real or complex numbers 15. Functions of a single real variable limits and continuity 16. Monotonic functions functions of bounded variation 17. Differentiation mean-value theorems 18. The nth mean-value theorem: Taylor's theorem 19. Convex and concave functions 20. The elementary transcendental functions 21. Inequalities 22 The Riemann integral 23. Integration and differentiation 24. The Riemann-Stiltjes integral 25. Improper integrals convergence of integrals 26. Further tests for the convergence of series 27. Uniform convergence 28. Functions of two real variables. Continuity and differentiability Hints on the solution of exercises and answers to exercises Appendix Index.
Preface Notation and Conventions Preliminaries 1. Enumerability and sequences 2. Bounds for sets of numbers 3. Bounds for functions and sequences 4. Limits of the sequences 5. Monotonic sequences two important examples irrational powers of positive real numbers 6. Upper and lower limits of real sequences: the general principle of convergence 7. Convergence of series absolute convergence 8. Conditional convergence 9. Rearrangement and multiplication of absolutely convergent series 10. Double series 11. Power series 12. Point set theory 13. The Bolzano-Weierstrass, Cantor and Heine-Borel theorems 14. Functions defined over real or complex numbers 15. Functions of a single real variable limits and continuity 16. Monotonic functions functions of bounded variation 17. Differentiation mean-value theorems 18. The nth mean-value theorem: Taylor's theorem 19. Convex and concave functions 20. The elementary transcendental functions 21. Inequalities 22 The Riemann integral 23. Integration and differentiation 24. The Riemann-Stiltjes integral 25. Improper integrals convergence of integrals 26. Further tests for the convergence of series 27. Uniform convergence 28. Functions of two real variables. Continuity and differentiability Hints on the solution of exercises and answers to exercises Appendix Index.
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