Problems in Real Analysis Teodora-Liliana T. Radulescu˘ Vicen¸tiuD. Radulescu˘ Titu Andreescu
Problems in Real Analysis
Advanced Calculus on the Real Axis Teodora-Liliana T. Radulescu˘ Vicen¸tiuD. Radulescu˘ Department of Mathematics Simion Stoilow Mathematics Institute Fratii Buzesti National College Romanian Academy Craiova 200352 Bucharest 014700 Romania Romania [email protected] [email protected]
Titu Andreescu School of Natural Sciences and Mathematics University of Texas at Dallas Richardson, TX 75080 USA [email protected]
ISBN: 978-0-387-77378-0 e-ISBN: 978-0-387-77379-7 DOI: 10.1007/978-0-387-77379-7 Springer Dordrecht Heidelberg London New York
Library of Congress Control Number: 2009926486
Mathematics Subject Classification (2000): 00A07, 26-01, 28-01, 40-01
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Springer is part of Springer Science+Business Media (www.springer.com) To understand mathematics means to be able to do mathematics. And what does it mean doing mathematics? In the first place it means to be able to solve mathematical problems. —George P´olya (1887–1985)
We come nearest to the great when we are great in humility. —Rabindranath Tagore (1861–1941) Foreword
This carefully written book presents an extremely motivating and original approach, by means of problem-solving, to calculus on the real line, and as such, serves as a perfect introduction to real analysis. To achieve their goal, the authors have care- fully selected problems that cover an impressive range of topics, all at the core of the subject. Some problems are genuinely difficult, but solving them will be highly rewarding, since each problem opens a new vista in the understanding of mathematics. This book is also perfect for self-study, since solutions are provided. I like the care with which the authors intersperse their text with careful reviews of the background material needed in each chapter, thought-provoking quotations, and highly interesting and well-documented historical notes. In short, this book also makes very pleasant reading, and I am confident that each of its readers will enjoy reading it as much as I did. The charm and never-ending beauty of mathematics pervade all its pages. In addition, this little gem illustrates the idea that one cannot learn mathematics without solving difficult problems. It is a world apart from the “computer addiction” that we are unfortunately witnessing among the younger generations of would-be mathematicians, who use too much ready-made software instead or their brains, or who stand in awe in front of computer-generated images, as if they had become the essence of mathematics. As such, it carries a very useful message. One cannot help comparing this book to a “great ancestor,” the famed Problems and Theorems in Analysis,byP´olya and Szeg˝o,a text that has strongly influenced generations of analysts. I am confident that this book will have a similar impact.
Hong Kong, July 2008 Philippe G. Ciarlet
vii Preface
If I have seen further it is by standing on the shoulders of giants. —Sir Isaac Newton (1642–1727), Letter to Robert Hooke, 1675
Mathematical analysis is central to mathematics, whether pure or applied. This discipline arises in various mathematical models whose dependent variables vary continuously and are functions of one or several variables. Real analysis dates to the mid-nineteenth century, and its roots go back to the pioneering papers by Cauchy, Riemann, and Weierstrass. In 1821, Cauchy established new requirements of rigor in his celebrated Cours d’Analyse. The questions he raised are the following: – What is a derivative really? Answer: a limit. – What is an integral really? Answer: a limit. – What is an infinite series really? Answer: a limit. This leads to – What is a limit? Answer: a number. And, finally, the last question: – What is a number? Weierstrass and his collaborators (Heine, Cantor) answered this question around 1870–1872. Our treatment in this volume is strongly related to the pioneering contributions in differential calculus by Newton, Leibniz, Descartes, and Euler in the seventeenth and eighteenth centuries, with mathematical rigor in the nineteenth century pro- moted by Cauchy, Weierstrass, and Peano. This presentation furthers modern directions in the integral calculus developed by Riemann and Darboux. Due to the huge impact of mathematical analysis, we have intended in this book to build a bridge between ordinary high-school or undergraduate exercises and more difficult and abstract concepts or problems related to this field. We present in this volume an unusual collection of creative problems in elementary mathematical anal- ysis. We intend to develop some basic principles and solution techniques and to offer a systematic illustration of how to organize the natural transition from problem- solving activity toward exploring, investigating, and discovering new results and properties.
ix x Preface
The aim of this volume in elementary mathematical analysis is to introduce, through problems-solving, fundamental ideas and methods without losing sight of the context in which they first developed and the role they play in science and partic- ularly in physics and other applied sciences. This volume aims at rapidly developing differential and integral calculus for real-valued functions of one real variable, giving relevance to the discussion of some differential equations and maximum prin- ciples. The book is mainly geared toward students studying the basic principles of math- ematical analysis. However, given its selection of problems, organization, and level, it would be an ideal choice for tutorial or problem-solving seminars, particularly those geared toward the Putnam exam and other high-level mathematical contests. We also address this work to motivated high-school and undergraduate students. This volume is meant primarily for students in mathematics, physics, engineering, and computer science, but, not without authorial ambition, we believe it can be used by anyone who wants to learn elementary mathematical analysis by solving prob- lems. The book is also a must-have for instructors wishing to enrich their teach- ing with some carefully chosen problems and for individuals who are interested in solving difficult problems in mathematical analysis on the real axis. The volume is intended as a challenge to involve students as active participants in the course. To make our work self-contained, all chapters include basic definitions and properties. The problems are clustered by topic into eight chapters, each of them containing both sections of proposed problems with complete solutions and separate sections including auxiliary problems, their solutions being left to our readers. Throughout the book, students are encouraged to express their own ideas, solutions, generaliza- tions, conjectures, and conclusions. The volume contains a comprehensive collection of challenging problems, our goal being twofold: first, to encourage the readers to move away from routine exercises and memorized algorithms toward creative solutions and nonstandard problem-solving techniques; and second, to help our readers to develop a host of new mathematical tools and strategies that will be useful beyond the classroom and in a number of applied disciplines. We include representative problems proposed at various national or international competitions, problems selected from prestigious mathematical journals, but also some original problems published in leading publi- cations. That is why most of the problems contained in this book are neither standard nor easy. The readers will find both classical topics of mathematical analysis on the real axis and modern ones. Additionally, historical comments and developments are presented throughout the book in order to stimulate further inquiry. Traditionally, a rigorous first course or problem book in elementary mathematical analysis progresses in the following order:
Sequences Functions =⇒ Continuity =⇒ Differentiability =⇒ Integration Limits Preface xi
However, the historical development of these subjects occurred in reverse order:
Archimedes Newton (1665) Cauchy (1821) ⇐= Weierstrass (1872) ⇐= ⇐= Kepler (1615) Leibniz (1675) Fermat (1638)
This book brings to life the connections among different areas of mathematical analysis and explains how various subject areas flow from one another. The vol- ume illustrates the richness of elementary mathematical analysis as one of the most classical fields in mathematics. The topic is revisited from the higher viewpoint of university mathematics, presenting a deeper understanding of familiar subjects and an introduction to new and exciting research fields, such as Ginzburg–Landau equa- tions, the maximum principle, singular differential and integral inequalities, and nonlinear differential equations. The volume is divided into four parts, ten chapters, and two appendices, as follows:
Part I. Sequences, Series, and Limits Chapter 1. Sequences Chapter 2. Series Chapter 3. Limits of Functions Part II. Qualitative Properties of Continuous and Differentiable Functions Chapter 4. Continuity Chapter 5. Differentiability Part III. Applications to Convex Functions and Optimization Chapter 6. Convex Functions Chapter 7. Inequalities and Extremum Problems Part IV. Antiderivatives, Riemann Integrability, and Applications Chapter 8. Antiderivatives Chapter 9. Riemann Integrability Chapter 10. Applications of the Integral Calculus Appendix A. Basic Elements of Set Theory Appendix B. Topology of the Real Line
Each chapter is divided into sections. Exercises, formulas, and figures are num- bered consecutively in each section, and we also indicate both the chapter and the section numbers. We have included at the beginning of chapters and sections quo- tations from the literature. They are intended to give the flavor of mathematics as a science with a long history. This book also contains a rich glossary and index, as well as a list of abbreviations and notation. xii Preface
Key features of this volume: – contains a collection of challenging problems in elementary mathematical analysis; – includes incisive explanations of every important idea and develops illuminating applications of many theorems, along with detailed solutions, suitable cross- references, specific how-to hints, and suggestions; – is self-contained and assumes only a basic knowledge but opens the path to com- petitive research in the field; – uses competition-like problems as a platform for training typical inventive skills; – develops basic valuable techniques for solving problems in mathematical ana- lysis on the real axis; – 38 carefully drawn figures support the understanding of analytic concepts; – includes interesting and valuable historical account of ideas and methods in analysis; – contains excellent bibliography, glossary, and index. The book has elementary prerequisites, and it is designed to be used for lecture courses on methodology of mathematical research or discovery in mathematics. This work is a first step toward developing connections between analysis and other math- ematical disciplines, as well as physics and engineering. The background the student needs to read this book is quite modest. Anyone with elementary knowledge in calculus is well-prepared for almost everything to be found here. Taking into account the rich introductory blurbs provided with each chapter, no particular prerequisites are necessary, even if a dose of mathematical so- phistication is needed. The book develops many results that are rarely seen, and even experienced readers are likely to find material that is challenging and informative. Our vision throughout this volume is closely inspired by the following words of George P´olya [90] (1945) on the role of problems and discovery in mathematics: Infallible rules of discovery leading to the solution of all possible mathematical problems would be more desirable than the philosopher’s stone, vainly sought by all alchemists. The first rule of discovery is to have brains and good luck. The second rule of discovery is to sit tight and wait till you get a bright idea. Those of us who have little luck and less brain sometimes sit for decades. The fact seems to be, as Poincare´ observed, it is the man, not the method, that solves the problem. Despite our best intentions, errors are sure to have slipped by us. Please let us know of any you find.
August 2008 Teodora-Liliana Radulescu˘ Vicent¸iu Radulescu˘ Titu Andreescu Acknowledgments
We acknowledge, with unreserved gratitude, the crucial role of Professors Cather- ine Bandle, Wladimir-Georges Boskoff, Louis Funar, Patrizia Pucci, Richard Stong, and Michel Willem, who encouraged us to write a problem book on this subject. Our colleague and friend Professor Dorin Andrica has been very interested in this project and suggested some appropriate problems for this volume. We warmly thank Professors Ioan S¸erdean and Marian Tetiva for their kind support and useful discus- sions. This volume was completed while Vicent¸iu R˘adulescu was visiting the Univer- sity of Ljubljana during July and September 2008 with a research position funded by the Slovenian Research Agency. He would like to thank Professor Duˇsan Repovˇs for the invitation and many constructive discussions. We thank Dr. Nicolae Constantinescu and Dr. Mirel Cos¸ulschi for the profes- sional drawing of figures contained in this book. We are greatly indebted to the anonymous referees for their careful reading of the manuscript and for numerous comments and suggestions. These precious con- structive remarks were very useful to us in the elaboration of the final version of this volume. We are grateful to Ann Kostant, Springer editorial director for mathematics, for her efficient and enthusiastic help, as well as for numerous suggestions related to previous versions of this book. Our special thanks go also to Laura Held and to the other members of the editorial technical staff of Springer New York for the excellent quality of their work. We are particularly grateful to copyeditor David Kramer for his guidance, thor- oughness and attention to detail. V. Radulescu ˘ acknowledges the support received from the Romanian Research Council CNCSIS under Grant 55/2008 “Sisteme diferent¸iale ˆın analiza neliniar˘as¸i aplicat¸ii.”
xiii Contents
Foreword ...... vii
Preface ...... ix
Acknowledgments ...... xiii
Abbreviations and Notation ...... xix
Part I Sequences, Series, and Limits
1 Sequences ...... 3 1.1 MainDefinitionsandBasicResults ...... 3 1.2 Introductory Problems ...... 7 1.3 RecurrentSequences...... 18 1.4 Qualitative Results ...... 30 1.5 Hardy’s and Carleman’s Inequalities ...... 45 1.6 IndependentStudyProblems...... 51
2 Series ...... 59 2.1 MainDefinitionsandBasicResults ...... 59 2.2 ElementaryProblems...... 66 2.3 ConvergentandDivergentSeries...... 73 2.4 Infinite Products ...... 86 2.5 Qualitative Results ...... 89 2.6 IndependentStudyProblems...... 110
3 Limits of Functions ...... 115 3.1 MainDefinitionsandBasicResults ...... 115 3.2 ComputingLimits...... 118 3.3 Qualitative Results ...... 124 3.4 IndependentStudyProblems...... 133
xv xvi Contents
Part II Qualitative Properties of Continuous and Differentiable Functions
4 Continuity ...... 139 4.1 TheConceptofContinuityandBasicProperties...... 139 4.2 ElementaryProblems...... 144 4.3 TheIntermediateValueProperty...... 147 4.4 Types of Discontinuities ...... 151 4.5 FixedPoints...... 154 4.6 Functional Equations and Inequalities ...... 163 4.7 Qualitative Properties of Continuous Functions ...... 169 4.8 IndependentStudyProblems...... 177
5 Differentiability ...... 183 5.1 TheConceptofDerivativeandBasicProperties...... 183 5.2 Introductory Problems ...... 198 5.3 TheMainTheorems...... 218 5.4 TheMaximumPrinciple...... 235 5.5 Differential Equations and Inequalities ...... 238 5.6 IndependentStudyProblems...... 252
Part III Applications to Convex Functions and Optimization
6 Convex Functions ...... 263 6.1 MainDefinitionsandBasicResults ...... 263 6.2 BasicPropertiesofConvexFunctionsandApplications...... 265 6.3 Convexity versus Continuity and Differentiability ...... 273 6.4 Qualitative Results ...... 278 6.5 IndependentStudyProblems...... 285
7 Inequalities and Extremum Problems ...... 289 7.1 BasicTools...... 289 7.2 ElementaryExamples...... 290 7.3 Jensen, Young, H¨older, Minkowski, and Beyond ...... 294 7.4 OptimizationProblems...... 300 7.5 Qualitative Results ...... 305 7.6 IndependentStudyProblems...... 308
Part IV Antiderivatives, Riemann Integrability, and Applications
8 Antiderivatives ...... 313 8.1 MainDefinitionsandProperties ...... 313 8.2 ElementaryExamples...... 315 8.3 ExistenceorNonexistenceofAntiderivatives...... 317 8.4 Qualitative Results ...... 319 8.5 IndependentStudyProblems...... 324 Contents xvii
9 Riemann Integrability ...... 325 9.1 MainDefinitionsandProperties ...... 325 9.2 ElementaryExamples...... 329 9.3 ClassesofRiemannIntegrableFunctions...... 337 9.4 BasicRulesforComputingIntegrals ...... 339 9.5 RiemannIintegralsandLimits...... 341 9.6 Qualitative Results ...... 351 9.7 IndependentStudyProblems...... 367
10 Applications of the Integral Calculus ...... 373 10.1Overview...... 373 10.2 Integral Inequalities...... 374 10.3ImproperIntegrals...... 390 10.4IntegralsandSeries...... 402 10.5ApplicationstoGeometry...... 406 10.6IndependentStudyProblems...... 409
Part V Appendix
A Basic Elements of Set Theory ...... 417 A.1 DirectandInverseImageofaSet...... 417 A.2 Finite, Countable, and Uncountable Sets ...... 418
B Topology of the Real Line ...... 419 B.1 OpenandClosedSets...... 419 B.2 Some Distinguished Points...... 420
Glossary ...... 421
References ...... 437
Index ...... 443 Abbreviations and Notation
Abbreviations We have tried to avoid using nonstandard abbreviations as much as possible. Other abbreviations include: AMM American Mathematical Monthly GMA Mathematics Gazette, Series A MM Mathematics Magazine IMO International Mathematical Olympiad IMCUS International Mathematics Competition for University Students MSC Mikl´os Schweitzer Competitions Putnam The William Lowell Putnam Mathematical Competition SEEMOUS South Eastern European Mathematical Olympiad for University Students
Notation We assume familiarity with standard elementary notation of set theory, logic, algebra, analysis, number theory, and combinatorics. The following is notation that deserves additional clarification. N the set of nonnegative integers (N = {0,1,2,3,...}) N∗ the set of positive integers (N∗ = {1,2,3,...}) Z the set of integer real numbers (Z = {...,−3,−2,−1,0,1,2,3,...}) Z∗ the set of nonzero integer real numbers (Z∗ = Z \{0}) Q the set of rational real numbers Q = m ∈ Z, ∈ N∗, n ; m n m and n are relatively prime R the set of real numbers R∗ the set of nonzero real numbers (R∗ = R \{0}) R+ the set of nonnegative real numbers (R+ =[0 ,+∞)) R the completed real line R = R ∪{−∞,+∞}
xix xx Abbreviations and Notation
C the set of complex numbers + 1 n = . ... e limn→∞ 1 n 2 71828 supA the least upper bound of the set A ⊂ R infA the greatest lower bound of the set A ⊂ R x+ the positive part of the real number x (x+ = max{x,0}) x− the negative part of the real number x (x− = max{−x,0}) |x| the modulus (absolute value) of the real number x (|x| = x+ + x−) {x} the fractional part of the real number x (x =[x]+{x}) Card(A) cardinality of the finite set A dist(x,A) the distance from x ∈ R to the set A ⊂ R (dist(x,A)=inf{|x − a|; a ∈ A}) IntA the set of interior points of A ⊂ R f (A) the image of the set A under a mapping f f −1(B) the inverse image of the set B under a mapping f f ◦ g the composition of functions f and g: ( f ◦ g)(x)= f (g(x)) n! n factorial, equal to n(n − 1)···1 (n ∈ N∗) (2n)!! 2n(2n − 2)(2n − 4)···4 · 2(n ∈ N∗) (2n + 1)!! (2n + 1)(2n − 1)(2n − 3)···3 · 1(n ∈ N∗) > lnx loge x (x 0) x x0 x→x0 ∈ R and x < x0 x x0 x→x0 ∈ R and x > x0 limsupxn lim supxk →∞ n→∞ n k≥n liminfxn lim inf xk n→∞ n→∞ k≥n f (n)(x) nth derivative of the function f at x Cn(a,b) the set of n-times differentiable functions f : (a,b)→R such that f (n) is continuous on (a,b) ∞( , ) ( , )→R C a b the set of infinitely differentiable functions f : a b ( ∞( , )= ∞ n( , )) C a b n=0 C a b N Δ f the Laplace operator applied to the function f : D ⊂ R →R ∂ 2 f ∂ 2 f ∂ 2 f Δ f = + + ···+ ∂ 2 ∂ 2 ∂ 2 x1 x2 xN Landau’s notation f (x)=o(g(x)) as x→x0 if f (x)/g(x)→0asx→x0 f (x)=O(g(x)) as x→x0 if f (x)/g(x) is bounded in a neighborhood of x0 f ∼ g as x→x0 if f (x)/g(x)→1asx→x0 Hardy’s notation f ≺≺ g as x→x0 if f (x)/g(x)→0asx→x0 f g as x→x0 if f (x)/g(x) is bounded in a neighborhood of x0 Chapter 1 Sequences
As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality. —Albert Einstein (1879–1955)
Abstract. In this chapter we study real sequences, a special class of functions whose domain is the set N of natural numbers and range a set of real numbers.
1.1 Main Definitions and Basic Results
Hypotheses non fingo. [“I frame no hypotheses.”] Sir Isaac Newton (1642–1727)
Sequences describe wide classes of discrete processes arising in various applica- tions. The theory of sequences is also viewed as a preliminary step in the attempt to model continuous phenomena in nature. Since ancient times, mathematicians have realized that it is difficult to reconcile the discrete with the continuous. We under- stand counting 1, 2, 3,... up to arbitrarily large numbers, but do we also under- stand moving from 0 to 1 through the continuum of points between them? Around 450 BC, Zeno thought not, because continuous motion involves infinity in an essential way. As he put it in his paradox of dichotomy: There is no motion because that which is moved must arrive at the middle (of its course) before it arrives at the end. Aristotle, Physics, Book VI, Ch. 9 A sequence of real numbers is a function f : N→R (or f : N∗→R). We usually write an (or bn, xn, etc.) instead of f (n).If(an)n≥1 is a sequence of real numbers and if n1 < n2 < ··· < nk < ··· is an increasing sequence of positive integers, then ( ) ( ) the sequence ank k≥1 is called a subsequence of an n≥1. A sequence of real numbers (an)n≥1 is said to be nondecreasing (resp., increas- ing)ifan ≤ an+1 (resp., an < an+1), for all n ≥ 1. The sequence (an)n≥1 is called nonincreasing (resp., decreasing) if the above inequalities hold with “≥” (resp., “>”) instead of “≤” (resp., “<”).
T.-L. Rădulescu et al., Problems in Real Analysis: Advanced Calculus on the Real Axis, 3 DOI: 10.1007/978-0-387-77379-7_1, _ © Springer Science + Business Media, LLC 2009 4 1 Sequences
y 1 y = l + ε
y = l − ε
0 123 N x Fig. 1.1 Adapted from G.H. Hardy, Pure Mathematics, Cambridge University Press, 1952.
We recall in what follows some basic definitions and properties related to sequences. One of the main notions we need in the sequel is that of convergence. Let (an)n≥1 be a sequence of real numbers. We say that (an)n≥1 is a convergent sequence if there exists ∈ R (which is called the limit of (an)n≥1) such that for each neighborhood N of ,wehavean ∈ N for all n ≥ N,whereN is a positive integer depending on N .Inotherwords,(an)n≥1 converges to if and only if for each ε > 0 there exists a natural number N = N(ε) such that |an − | < ε,forall n ≥ N. In this case we write limn→∞ an = or an→ as n→∞. The concentration of an’s in the strip ( − ε, + ε) for n ≥ N is depicted in Figure 1.1. =( + / )n =( + / )n+1 = ∑n / Example. Define an 1 1 n , bn 1 1 n , cn k=0 1 k!. Then (an)n≥1 and (cn)n≥1 are increasing sequences, while (bn)n≥1 is a decreasing seq- uence. However, all these three sequences have the same limit, which is denoted by e(e= 2.71828...). We will prove in the next chapter that e is an irrational number. Another important example is given by the following√ formula due to Stirling, which asserts that, asymptotically, n! behaves like nne−n 2π n. More precisely,
n! lim √ = 1. n→∞ nne−n 2π n Stirling’s formula is important in calculating limits, because without this asymptotic property it is difficult to estimate the size of n!forlargen. In this capacity, it plays an important role in probability theory, when it is used in computing the probable outcome of an event after a very large number of trials. We refer to Chapter 9 for a complete proof of Stirling’s formula. We say that the sequence of real numbers (an)n≥1 has limit +∞ (resp., −∞)if to each α ∈ R there corresponds some positive integer N0 such that n ≥ N0 implies an > α (resp., an < α). We do not say that (an)n≥1 converges in these cases. An element α ∈ R is called an accumulation point of the sequence (an)n≥1 ⊂ R ( ) →α →∞ if there exists a subsequence ank k≥1 such that ank as nk . Any convergent sequence is bounded. The converse is not true (find a counter- example!), but we still have a partial positive answer, which is contained in the following result, which is due to Bernard Bolzano (1781–1848) and 1.1 Main Definitions and Basic Results 5
Karl Weierstrass (1815–1897). It seems that this theorem was revealed for the first time in Weierstrass’s lecture of 1874. Bolzano–Weierstrass Theorem. A bounded sequence of real numbers has a convergent subsequence. However, some additional assumptions can guarantee that a bounded sequence converges. An important criterion for convergence is stated in what follows. Monotone Convergence Theorem. Let (an)n≥1 be a bounded sequence that is monotone. Then (an)n≥1 is a convergent sequence. If increasing, then limn→∞ an = = supn an, and if decreasing, then limn→∞ an infn an. In many arguments a central role is played by the following principle due to Cantor. Nested Intervals Theorem. Suppose that In =[an,bn] are closed intervals such that In+1 ⊂ In,foralln ≥ 1.Iflimn→∞(bn − an)=0, then there is a unique real number x0 that belongs to every In. A sequence (an)n≥1 of real numbers is called a Cauchy sequence [Augustin Louis Cauchy (1789–1857)] if for every ε > 0 there is a natural number Nε such that |am − an| < ε,forallm,n ≥ Nε . A useful result is that any Cauchy sequence is a bounded sequence. The following strong convergence criterion reduces the study of convergent sequences to that of Cauchy sequences. Cauchy’s Criterion. A sequence of real numbers is convergent if and only if it is a Cauchy sequence. Let (an)n≥1 be an arbitrary sequence of real numbers. The limit inferior of (an)n≥1 (denoted by liminfn→∞ an) is the supremum of the set X of x ∈ R such ∗ that there are at most a finite numbers of n ∈ N for which an < x.Thelimit su- ( ) ∈ R perior of an n≥1 (denoted by limsupn→∞ an) is the infimum of the set Y of y such that there are finitely many positive integers n for which an > y. Equivalent characterizations of these notions are the following: = ∈ R α <