An Introduction to Real Analysis John K. Hunter

1 An Introduction to real AnalysisJohn K. Hunter1 Department of Mathematics, University of California at Davis1 The author was supported in part by the NSF. Thanks to Janko Gravner for a number of correc-tions and are some notes on introductory real Analysis . They coverthe properties of the real numbers, sequences and series of real numbers, limitsof functions, continuity, differentiability, sequences and series of functions, andRiemann integration. They don t include multi-variable calculus or containany problem sets. Optional sections are John K. Hunter , 2014 ContentsChapter 1. Sets and Composition and inverses of Indexed Countable and uncountable sets14 Chapter 2. Rational real numbers: algebraic real numbers: ordering The supremum and real numbers: Properties of the supremum and infimum31 Chapter 3. The absolute Convergence and Properties of Monotone The lim sup and lim Cauchy The Bolzano-Weierstrass theorem57 Chapter 4.

2 Convergence of The Cauchy Absolutely convergent The comparison * The Riemann The ratio and root Alternating The Cauchy * Double * The irrationality ofe86 Chapter 5. Topology of the real Open Closed Compact Connected * The Cantor set104 Chapter 6. Limits of Left, right, and infinite Properties of limits117 Chapter 7. Continuous Properties of continuous Uniform Continuous functions and open Continuous functions on compact The intermediate value Monotonic functions136 Chapter 8. Differentiable The Properties of the The chain Extreme The mean value Taylor s * The inverse function * L H ospital s rule162 Chapter 9. Sequences and Series of Pointwise Uniform Cauchy condition for uniform Properties of uniform Series175 Chapter 10.

3 Power Radius of Examples of power Algebraic operations on power Differentiation of power The exponential * Smooth versus analytic functions197 Chapter 11. The Riemann The supremum and infimum of Definition of the The Cauchy criterion for Continuous and monotonic Linearity, monotonicity, and Further existence * Riemann * The Lebesgue criterion238 Chapter 12. Properties and Applications of the The fundamental theorem of Consequences of the fundamental Integrals and sequences of Improper Riemann * Principal value The integral test for Taylor s theorem with integral remainder268 Chapter 13. Metric, Normed, and Topological Metric Normed Open and closed Completeness, compactness, and Topological * Function * The Minkowski inequality293 Bibliography299 Chapter 1 Sets and FunctionsWe understand a set to be any collectionMof certain distinct objectsof our thought or intuition (called the elements ofM) into a whole.

4 (Georg Cantor, 1895)In mathematics you don t understand things. You just get used to them.(Attributed to John von Neumann)In this chapter, we define sets, functions, and relations and discuss some oftheir general properties. This material can be referred back to as needed in thesubsequent SetsA set is a collection of objects, called the elements or members of the set. Theobjects could be anything (planets, squirrels, characters in Shakespeare s plays, orother sets) but for us they will be mathematical objects such as numbers, or setsof numbers. We writex Xifxis an element of the setXandx / Xifxis notan element the definition of a set as a collection seems circular, that s because itis. Conceiving of many objects as a single whole is a basic intuition that cannotbe analyzed further, and the the notions of set and membership are primitiveones.

5 These notions can be made mathematically precise by introducing a systemof axioms for sets and membership that agrees with our intuition and proving otherset-theoretic properties from the most commonly used axioms for sets are the ZFC axioms, named somewhatinconsistently after two of their founders (Zermelo and Fraenkel) and one of theiraxioms (the Axiom of Choice). We won t state these axioms here; instead, we use naive set theory, based on the intuitive properties of sets. Nevertheless, all theset-theory arguments we use can be rigorously formalized within the ZFC Sets and FunctionsSets are determined entirely by their elements. Thus, the setsX,Yare equal,writtenX=Y, ifx Xif and only ifx is convenient to define the empty set, denoted by , as the set with no elements.(Since sets are determined by their elements, there is only one set with no elements!)IfX6= , meaning thatXhas at least one element, then we say thatXis can define a finite set by listing its elements (between curly brackets).

6 Forexample,X={2,3,5,7,11}is a set with five elements. The order in which the elements are listed or repetitionsof the same element are irrelevant. Alternatively, we can defineXas the set whoseelements are the first five prime numbers. It doesn t matter how we specify theelements ofX, only that they are the sets can t be defined by explicitly listing all of their elements. Never-theless, we will adopt a realist (or platonist ) approach towards arbitrary infinitesets and regard them as well-defined totalities. In constructive mathematics andcomputer science, one may be interested only in sets that can be defined by a rule oralgorithm for example, the set of all prime numbers rather than by infinitelymany arbitrary specifications, and there are some mathematicians who considerinfinite sets to be meaningless without some way of constructing them. Similarissues arise with the notion of arbitrary subsets, functions, and infinite sets we use are derived from the natural and realnumbers, about which we have a direct intuitive understanding of the natural numbers 1,2,3.

7 Derives from denote the set of natural numbers byN={1,2,3,..}.We defineNso that it starts at 1. In set theory and logic, the natural numbersare defined to start at zero, but we denote this set byN0={0,1,2,..}. Histori-cally, the number 0 was later addition to the number system, primarily by Indianmathematicians in the 5th century AD. The ancient Greek mathematicians, suchas Euclid, defined a number as a multiplicity and didn t consider 1 to be a understanding of the real numbers derives from durations of time andlengths in space. We think of the real line, or continuum, as being composed of an(uncountably) infinite number of points, each of which corresponds to a real number,and denote the set of real numbers byR. There are philosophical questions, goingback at least to Zeno s paradoxes, about whether the continuum can be representedas a set of points, and a number of mathematicians have disputed this assumptionor introduced alternative models of the continuum.

8 There are, however, no knowninconsistencies in treatingRas a set of points, and since Cantor s work it has beenthe dominant point of view in mathematics because of its precision, power, Sets3We denote the set of (positive, negative and zero) integers byZ={.., 3, 2, 1,0,1,2,3,..},and the set of rational numbers (ratios of integers) byQ={p/q:p,q Zandq6= 0}.The letter Z comes from zahl (German for number ) and Q comes from quotient. These number systems are discussed further in Chapter we will not develop any complex Analysis here, we occasionally makeuse of complex numbers. We denote the set of complex numbers byC={x+iy:x,y R},where we add and multiply complex numbers in the natural way, with the additionalidentity thati2= 1, meaning thatiis a square root of 1. Ifz=x+iy C, wecallx=<zthe real part ofzandy==zthe imaginary part ofz, and we call|z|= x2+y2the absolute value, or modulus, ofz.

9 Two complex numbersz=x+iy,w=u+ivare equal if and only ifx=uandy= setAis a subset of a setX, writtenA XorX A, ifevery element ofAbelongs toX; that is, ifx Aimplies thatx also say thatAis included example, ifPis the set of prime numbers,thenP N, andN R. The empty set and the whole setXare subsets of anysetX. Note thatX=Yif and only ifX YandY X; we often prove theequality of two sets by showing that each one includes the our notation,A Xdoes not imply thatAis a proper subset ofX(thatis, a subset ofXnot equal toXitself), and we may haveA=X. This notationfor non-strict inclusion is not universal; some authors useA Xto denote strictinclusion, in whichA6=X, andA Xto denote non-strict inclusion, in whichA=Xis power setP(X) of a setXis the set of all subsets {1,2,3}, thenP(X) ={ ,{1},{2},{3},{2,3},{1,3},{1,2},{1,2,3}}. The power set of a finite set withnelements has 2nelements because, indefining a subset, we have two independent choices for each element (does it belongto the subset or not?)

10 In Example ,Xhas 3 elements andP(X) has 23= power set of an infinite set, such asN, consists of all finite and infinitesubsets and is infinite. We can define finite subsets ofN, or subsets with finite1By contrast, we say that an elementx XiscontainedinX, in which cases the singleton set{x}isincludedinX. This terminological distinction is not universal, but it is almost always clear fromthe context whether one is referring to an element of a set or a subset of a set. In fact, before thedevelopment of the contemporary notation for set theory, Dedekind [3] used the same symbol ( ) todenote both membership of elements and inclusion of Sets and Functionscomplements, by listing finitely many elements. Some infinite subsets, such asthe set of primes or the set of squares, can be defined by giving a definite rulefor membership.