Elementary Real Analysis

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.