Mathematics A Course in Real Analysis provides a rigorous treatment of the foundations of differential and integral calculus at the advanced undergraduate level The third part consists of appendices on set theory and linear algebra as well as solutions to some of the exercises Features • Provides a detailed axiomatic account of the real number system • Develops the Lebesgue integral on n from the beginning • Gives an in-depth description of the algebra and calculus of differential forms on surfaces in n • Offers an easy transition to the more advanced setting of differentiable manifolds by covering proofs of Stokes’s theorem and the divergence theorem at the concrete level of compact surfaces in n • Summarizes relevant results from elementary set theory and linear algebra • Contains over 90 figures that illustrate the essential ideas behind a concept or proof • Includes more than 1,600 exercises throughout the text, with selected solutions in an appendix • Access online or download to your smartphone, tablet or PC/Mac • Search the full text of this and other titles you own • Make and share notes and highlights • Copy and paste text and figures for use in your own documents • Customize your view by changing font size and layout Tai ngay!!! Ban co the xoa dong chu nay!!! K22153 w w w c rc p r e s s c o m JUNGHENN With clear proofs, detailed examples, and numerous exercises, this book gives a thorough treatment of the subject It progresses from single variable to multivariable functions, providing a logical development of material that will prepare readers for more advanced analysis-based studies A COURSE IN The second part focuses on functions of several variables It introduces the topological ideas needed (such as compact and connected sets) to describe analytical properties of multivariable functions This part also discusses differentiability and integrability of multivariable functions and develops the theory of differential forms on surfaces in n REAL ANALYSIS The first part of the text presents the calculus of functions of one variable This part covers traditional topics, such as sequences, continuity, differentiability, Riemann integrability, numerical series, and the convergence of sequences and series of functions It also includes optional sections on Stirling’s formula, functions of bounded variation, Riemann–Stieltjes integration, and other topics WITH VITALSOURCE ® EBOOK A COURSE IN REAL ANALYSIS HUGO D JUNGHENN A COURSE IN REAL ANALYSIS K22153_FM.indd 1/9/15 4:46 PM K22153_FM.indd 1/9/15 4:46 PM A COURSE IN REAL ANALYSIS HUGO D JUNGHENN The George Washington University Washington, D.C., USA K22153_FM.indd 1/9/15 4:46 PM CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2015 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S Government works Version Date: 20150109 International Standard Book Number-13: 978-1-4822-1928-9 (eBook - PDF) This book contains information obtained from authentic and highly regarded sources Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint Except as permitted under U.S Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400 CCC is a not-for-profit organization that provides licenses and registration for a variety of users For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com TO THE MEMORY OF MY PARENTS Rita and Hugo Contents Preface xi List of Figures xiii List of Tables xvii List of Symbols I xix Functions of One Variable 1 The Real Number System 1.1 From Natural Numbers to Real Numbers 1.2 Algebraic Properties of R 1.3 Order Structure of R 1.4 Completeness Property of R 1.5 Mathematical Induction 1.6 Euclidean Space 3 12 19 24 Numerical Sequences 2.1 Limits of Sequences 2.2 Monotone Sequences 2.3 Subsequences and Cauchy Sequences 2.4 Limits Inferior and Superior 29 29 36 38 42 47 47 55 59 63 67 73 73 80 85 88 94 Limits and Continuity on R 3.1 Limit of a Function *3.2 Limits Inferior and Superior 3.3 Continuous Functions 3.4 Properties of Continuous Functions 3.5 Uniform Continuity Differentiation on R 4.1 Definition of Derivative and Examples 4.2 The Mean Value Theorem *4.3 Convex Functions 4.4 Inverse Functions 4.5 L’Hospital’s Rule vii viii Contents 4.6 *4.7 Taylor’s Theorem on R Newton’s Method Riemann Integration on R 5.1 The Riemann–Darboux Integral 5.2 Properties of the Integral 5.3 Evaluation of the Integral *5.4 Stirling’s Formula 5.5 Integral Mean Value Theorems *5.6 Estimation of the Integral 5.7 Improper Integrals 5.8 A Deeper Look at Riemann Integrability *5.9 Functions of Bounded Variation *5.10 The Riemann–Stieltjes Integral Numerical Infinite Series 6.1 Definition and Examples 6.2 Series with Nonnegative Terms 6.3 More Refined Convergence Tests 6.4 Absolute and Conditional Convergence *6.5 Double Sequences and Series 107 107 116 120 129 131 134 143 151 152 156 163 163 169 176 181 188 Sequences and Series of Functions 7.1 Convergence of Sequences of Functions 7.2 Properties of the Limit Function 7.3 Convergence of Series of Functions 7.4 Power Series II Functions of Several Variables Metric Spaces 8.1 Definitions and Examples 8.2 Open and Closed Sets 8.3 Closure, Interior, and Boundary 8.4 Limits and Continuity 8.5 Compact Sets *8.6 The Arzelà–Ascoli Theorem 8.7 Connected Sets 8.8 The Stone–Weierstrass Theorem *8.9 Baire’s Theorem 100 103 193 193 199 204 211 229 231 231 238 243 248 255 263 268 275 282 Differentiation on Rn 9.1 Definition of the Derivative 9.2 Properties of the Differential 9.3 Further Properties of the Differential 9.4 Inverse Function Theorem 287 287 295 301 306 Contents 9.5 9.6 9.7 *9.8 ix Implicit Function Theorem Higher Order Partial Derivatives Higher Order Differentials and Taylor’s Optimization 10 Lebesgue Measure on Rn 10.1 General Measure Theory 10.2 Lebesgue Outer Measure 10.3 Lebesgue Measure 10.4 Borel Sets 10.5 Measurable Functions Theorem 312 318 323 330 343 343 347 351 356 360 11 Lebesgue Integration on Rn 11.1 Riemann Integration on Rn 11.2 The Lebesgue Integral 11.3 Convergence Theorems 11.4 Connections with Riemann Integration 11.5 Iterated Integrals 11.6 Change of Variables 367 367 368 379 385 388 398 12 Curves and Surfaces in Rn 12.1 Parameterized Curves 12.2 Integration on Curves 12.3 Parameterized Surfaces 12.4 m-Dimensional Surfaces 13 Integration on Surfaces 13.1 Differential Forms 13.2 Integrals on Parameterized Surfaces 13.3 Partitions of Unity 13.4 Integration on Compact m-Surfaces 13.5 The Fundamental Theorems of Calculus *13.6 Closed Forms in Rn 447 447 461 472 475 478 495 409 409 412 422 432 III Appendices 503 A Set Theory 505 B Linear Algebra 509 C Solutions to Selected Problems 517 Bibliography 581 Index 583 < ε/2 + ε/2 = ε a c This establishes the existence of Rb a f dw as well as the desired equality 5.10.6 Example Consider the floor function integrator R n w(x) = bxc A slight modification of the argument in 5.10.2 shows that f (x) dbxc exists iff f is Rk left continuous at the integers 1, 2, , n, in which case k−1 f (x) dbxc = f (k) For such a function, 5.10.5 implies that Z n n Z k n X X f (x) dbxc = f (x) dbxc = f (k) ♦ k=1 k−1 The preceding example suggests that improper Riemann-Stieltjes integration could be used to provide a unified theory that includes both improper Riemann integrals and infinite series This is indeed possible; however, it turns out that Lebesgue integration is a more efficient approach Lebesgue theory on Rn is developed in Chapter 11 The following theorem reveals a remarkable symmetry between integrand and integrator 5.10.7 Integration by Parts Formula If f ∈ Rba (w), then w ∈ Rba (f ) and Z b Z b f dw + w df = f (b)w(b) − f (a)w(a) a a Proof For any partition P{x0 = a, x1 , , xn−1 , xn = b}, f (b)w(b) − f (a)w(a) = Sf (w, P, ξ) = n X j=1 n X j=1 f (xj )w(xj ) − w(ξj )f (xj ) − n X j=1 n X f (xj−1 )w(xj−1 ) and w(ξj )f (xj−1 ) j=1 Subtracting we obtain f (b)w(b) − f (a)w(a) − Sf (w, P, ξ) n n X X = f (xj−1 )[w(ξj ) − w(xj−1 )] + f (xj )[w(xj ) − w(ξj )] j=1 = Sw (f, Q, ζ), j=1 Riemann Integration on R 159 where ζ = (a, x1 , x1 , x2 , x2 , , xn−1 , xn−1 , b) and Q is the refinement of P obtained by adding the coordinates of ξ to P Therefore, ξ P a ξ1 ξ2 ξ3 x2 x1 ξ4 ξ5 x4 b ξ4 x4 ξ5 b x3 ζ Q a ξ1 x1 ξ2 x2 ξ3 x3 FIGURE 5.10: The partition Q Z b Z b (b)w(b) − f (a)w(a) − f dw − S (w, P, ξ) = (f, Q, ζ) − f dw f