Elementary Real Analysis

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.

H. G. Eggleston (Author)

9780521098687, Cambridge University Press

Paperback / softback, published 18 December 2008

296 pages
21.6 x 14 x 1.7 cm, 0.37 kg

This textbook covers all the theoretical aspects of real variable analysis which undergraduates reading mathematics are likely to require during the first two or three years of their course. It is based on lecture courses which the author has given in the universities of Wales, Cambridge and London. The subject is presented rigorously and without padding. Definitions are stated explicitly and the whole development of the subject is logical and self-contained. Complex numbers are used but the complex variable calculus is not. 'Applied analysis', such as differential equations and Fourier series, is not dealt with. 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.

Subject Areas: Mathematics [PB]