Introduction to Mathematical Analysis
I have tried to provide an introduction, at an elementary level, to some of the important topics in real analysis, without avoiding reference to the central role which the completeness of the real numbers plays throughout. Many elementary textbooks are wr
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Linear Equations Sequences and Series Differential Calculus Elementary Differential Equations and Operators Partial Derivatives Complex Numbers Principles of Dynamics Electrical and Mechanical Oscillations Vibrating Systems Vibrating Strings Fourier Series Solutions of Laplace's Equation Solid Geometry Numerical Approximation Integral Calculus Sets and Groups Differential Geometry Probability Theory Multiple Integrals Fourier and Laplace Transforms Introduction to Abstract Algebra Functions of a Complex Variable, 2 Vols Linear Programming Sets l!l1d Numbers
P. M. Cohn
J. A. Green
P. J. Hilton
G. E. H. Reuter P. J. Hilton W.Ledermann M. B. Glauert D. S. Jones R. F. Chisnell D. R. Bland I. N. Sneddon D. R. Bland P. M. Cohn B. R. Morton W. Ledermann J. A. Green K. L. Wardle A. M. Arthurs W. Ledermann P. D. Robinson C. R. J. Clapham D. O. Tall Kathleen Trustrum S. 8wierczkoswki
INTRODUCTION TO MATHEMATICAL ANALYSIS BY
C. R. J. CLAPHAM Department of Mathematics University of Aberdeen
ROUTLEDGE & KEGAN PAUL LONDON AND BOSTON
First published in 1973 by Routledge & Kegan Paul Ltd, Broadway House, 68-74 Carter Lane, London EC4V 5EL and 9 Park Street, Boston, Mass. 02108, U.S.A. Willmer Brothers Limited, Birkenhead
© C. R. J. Clapham, 1973 No part of this book may be reproduced in any form without permission from the publisher, except for the quotation of brief passages in criticism ISBN-13: 978-0-7100-7529-1
e-ISBN-13: 978-94-011-6572-3
DOl: 10.1007/978-94-011-6572-3
Library of Congress Catalog Card No. 72-95122
Contents
Preface
page vii
1. Axioms for the Real Numbers 1 2 3 4 5 6
Introduction Fields Order Completeness Upper bound The Archimedean property Exercises
1 1 6 9 11 13 15
2. Sequences 7 Limit of a sequence 8 Sequences without limits 9 Monotone sequences Exercises
18 22 23 25
3. Series 10 11 12 13 14
Infinite series Convergence Tests Absolute convergence Power series Exercises
27 27 29 31 32 34
4. Continuous Functions 15 16 17 18
Limit of a function Continuity The intermediate value property Bounds of a continuous function Exercises
36 37 40 41 43
5. Differentiable Functions 19 Derivatives 20 Rolle's theorem 21 The mean value theorem Exercises
45 47 49 51
6. The Riemann Integral 22 23 24 25 26 27 28 29 30 31 32
Introduction Upper and lower sums Riemann-integrable functions Examples A necessary and sufficient condition Monotone functions Uniform continuity Integrability of continuous functions Properties of the Riemann integral The mean value theorem Integration and differentiation Exercises
54 56 57 60 62 63 65 67 68 72 73 76
Answers to the Exercises
78
Index
81
Preface
I have tried to provide an introduction, at an elementary level, to some of the important topics in real analysis, without avoiding reference to the central role which the completeness of the real numbers plays throughout. Many elementary textbooks are written on the assumption that an appeal to the completeness axiom is beyond their scope; my aim here has been to give an account of the development from axiom
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