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An Introduction to Real Analysis John K. Hunter

An Introduction to real of Mathematics, University of California at are some notes on introductory real Analysis . They coverlimits of functions, continuity, differentiability, and sequences and series offunctions, but not Riemann integration A background in sequences and seriesof real numbers and some elementary point set topology of the real numbersis assumed, although some of this material is briefly john K. Hunter , 2012 ContentsChapter 1. The real Completeness Open Closed Accumulation points and isolated Compact sets7 Chapter 2.

An Introduction to Real Analysis John K. Hunter Department of Mathematics, University of California at Davis. Abstract. These are some notes on introductory real analysis. They cover limits of functions, continuity, differentiability, and sequences and series of ... real

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Transcription of An Introduction to Real Analysis John K. Hunter

1 An Introduction to real of Mathematics, University of California at are some notes on introductory real Analysis . They coverlimits of functions, continuity, differentiability, and sequences and series offunctions, but not Riemann integration A background in sequences and seriesof real numbers and some elementary point set topology of the real numbersis assumed, although some of this material is briefly john K. Hunter , 2012 ContentsChapter 1. The real Completeness Open Closed Accumulation points and isolated Compact sets7 Chapter 2.

2 Limits of Left, right, and infinite Properties of limits16 Chapter 3. Continuous Properties of continuous Uniform Continuous functions and open Continuous functions on compact The intermediate value Monotonic functions35 Chapter 4. Differentiable The Properties of the Extreme The mean value Taylor s theorem53 Chapter 5. Sequences and Series of Pointwise Uniform Cauchy condition for uniform Properties of uniform The The Spaces of continuous functions70 Chapter 6. Power Radius of Examples of power Differentiation of power The exponential Taylor s theorem and power Appendix: Review of series89 Chapter 7.

3 Metric Continuous Appendix: The Minkowski inequality102 Chapter 1 The real NumbersIn this chapter, we review some properties of the real numbersRand its don t give proofs for most of the results stated Completeness ofRIntuitively, unlike the rational numbersQ, the real numbersRform a continuumwith no gaps. There are two main ways to state this completeness, one in termsof the existence of suprema and the other in terms of the convergence of Suprema and nition Rbe a set of real numbers. A real numberM Ris anupper bound ofAifx Mfor everyx A, andm Ris a lower bound ofAifx mfor everyx A.

4 A set is bounded from above if it has an upper bound,bounded from below if it has a lower bound, and bounded if it has both an upperand a lower boundAn equivalent condition forAto be bounded is that there existsR Rsuchthat|x| Rfor everyx set of natural numbersN={1,2,3,4,..}is bounded from below by anym Rwithm 1. It is not bounded from above,soNis nition thatA Ris a set of real numbers. IfM Ris anupper bound ofAsuch thatM M for every upper boundM ofA, thenMiscalled the supremum or least upper bound ofA, denotedM= The real NumbersIfm Ris a lower bound ofAsuch thatm m for every lower boundm ofA,thenmis called the infimum or greatest lower bound ofA, denotedm= supremum or infimum of a set may or may not belong to the set.

5 IfsupA Adoes belong toA, then we also denote it by maxAand refer to it as themaximum ofA; if infA Athen we also denote it by minAand refer to it as theminimum finite set of real numbersA={x1,x2,..,xn}is bounded. Its supremum is the greatest element,supA= max{x1,x2,..,xn},and its infimum is the smallest element,infA= min{x1,x2,..,xn}.Both the supremum and infimum of a finite set belong to the {1n:n N}be the set of reciprocals of the natural numbers. Then supA= 1, which belongs toA, and infA= 0, which does not belong (0,1), we havesup(0,1) = 1,inf(0,1) = this case, neither supAnor infAbelongs toA.

6 The closed intervalB= [0,1],and the half-open intervalC= (0,1] have the same supremum and infimum supBand infBbelong toB, while only supCbelongs completeness ofRmay be expressed in terms of the existence of nonempty set of real numbers that is bounded from abovehas a infA= sup( A), it follows immediately that every nonempty set ofreal numbers that is bounded from below has an supremum of the set of real numbersA={x R:x < 2}is supA= 2. By contrast, since 2 is irrational, the set of rational numbersB={x Q:x < 2}has no supremum inQ. (IfM Qis an upper bound ofB, then there existsM Qwith 2< M < M, soMis not a least upper bound.))

7 Open Cauchy assume familiarity with the convergence of realsequences, but we recall the definition of Cauchy sequences and their relation withthe completeness nition sequence (xn) of real numbers is a Cauchy sequence if for every >0 there existsN Nsuch that|xm xn|< for allm,n > convergent sequence is Cauchy. Conversely, it follows from every Cauchy sequence of real numbers has a sequence of real numbers converges if and only if it is a fact that real Cauchy sequences have a limit is an equivalent way to formu-late the completeness ofR.

8 By contrast, the rational numbersQare not (xn) be a sequence of rational numbers such thatxn 2asn . Then (xn) is Cauchy inQbut (xn) does not have a limit Open setsOpen sets are among the most important subsets ofR. A collection of open sets iscalled a topology, and any property (such as compactness or continuity) that canbe defined entirely in terms of open sets is called a topological nition setG Ris open inRif for everyx Gthere exists a >0such thatG (x ,x+ ).Another way to state this definition is in terms of interior nition Rbe a subset ofR.

9 A pointx Ais an interior pointofAa if there is a >0 such thatA (x ,x+ ). A pointx Ris a boundarypoint ofAif every interval (x ,x+ ) contains points inAand points not , a set is open if and only if every point in the set is an interior open intervalI= (0,1) is open. Ifx IthenIcontains anopen interval aboutx,I (x2,1 +x2), x (x2,1 +x2),and, for example,I (x ,x+ ) if = min(x2,1 x2)> , every finite or infinite open interval (a,b), ( ,b), (a, ) is arbitrary union of open sets is open; one can prove that every open set inRis a countable union of disjoint open intervals.

10 A niteintersection of open setsis open, but an intersection of infinitely many open sets needn t be The real NumbersExample intervalIn=( 1n,1n)is open for everyn N, but n=1In={0}is not of using intervals to define open sets, we can use neighborhoods, and itis frequently simpler to refer to neighborhoods instead of open intervals of radius > nition setU Ris a neighborhood of a pointx RifU (x ,x+ )for some >0. The open interval (x ,x+ ) is called a -neighborhood neighborhood ofxneedn t be an open interval aboutx, it just has to containone.


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