Chapter2

1 Chapter 2 The supremum and infimumWe review the definition of the supremum and and infimum and some of theirproperties that we use in defining and analyzing the Riemann DefinitionFirst, we define upper and lower setA Rof real numbers is bounded from above if there existsa real numberM R, called an upper bound ofA, such thatx Mfor everyx A. Similarly,Ais bounded from below if there existsm R, called a lowerbound ofA, such thatx mfor everyx A. A set is bounded if it is boundedboth from above and supremum of a set is its least upper bound and the infimum is its greatestupper thatA Ris a set of real numbers. IfM Ris anupper bound ofAsuch thatM M for every upper boundM ofA, thenMiscalled the supremum ofA, denotedM= supA.

2 Ifm Ris a lower bound ofAsuch thatm m for every lower boundm ofA, thenmis called the or infimumofA, denotedm= not bounded from above, then we write supA= , and ifAis notbounded from below, we write infA= . IfA= is the empty set, then everyreal number is both an upper and a lower bound ofA, and we write sup = ,inf = . We will only say the supremum or infimum of a set exists if it is a finitereal number. For an indexed setA={xk:k J}, we often writesupA= supk Jxk,infA= infk supremum or infimum of a setAis unique if it , if both exist, then infA The supremum and thatM,M are suprema ofA. ThenM M sinceM is anupper bound ofAandMis a least upper bound; similarly,M M, soM=M.

3 Ifm,m are infima ofA, thenm m sincem is a lower bound ofAandmis agreatest lower bound; similarly,m m, som=m .If infAand supAexist, thenAis nonempty. Choosex A, TheninfA x supAsince infAis a lower bound ofAand supAis an upper bound. It follows thatinfA supA. If supA A, then we also denote it by maxAand call it the maximum ofA,and if infA A, then we also denote it by minAand call it the minimum {1/n:n N}. Then supA= 1 belongs toA, so maxA=1. On the other hand, infA= 0 doesn t belong toAandAhas no following alternative characterization of the sup and inf is an immediateconsequence of the R, thenM= supAif and only if: (a)Mis an upperbound ofA; (b) for everyM < Mthere existsx Asuch thatx > M.

4 Similarly,m= infAif and only if: (a)mis a lower bound ofA; (b) for everym > mthereexistsx Asuch thatx < m . the conditions in the proposition. ThenMis an upperbound and (b) implies that ifM < M, thenM is not an an upper bound, soM= supA. Conversely, ifM= supA, thenMis an upper bound, and ifM < MthenM is not an upper bound, so there existsx Asuch thatx > M . The prooffor the infimum is analogous. We frequently use one of the following arguments: (a) IfMis an upper bound ofA, thenM supA; (b) For every >0, there existsx Asuch thatx >supA .Similarly: (a) Ifmis an lower bound ofA, thenm infA; (b) For every >0,there existsx Asuch thatx <infA+ .The completeness of the real numbers ensures the existence of suprema andinfima.

5 In fact, the existence of suprema and infima is one way to define thecompleteness nonempty set of real numbers that is bounded from abovehas a supremum, and every nonempty set of real numbers that is bounded frombelow has an theorem is the basis of many existence results in real analysis. For exam-ple, once we show that a set is bounded from above, we can assert the existence ofa supremum without having to know its actual PropertiesIfA Randc R, then we definecA={y R:y=cxfor somex A}. Properties59 Proposition 0, thensupcA=csupA,infcA= <0, thensupcA=cinfA,infcA= result is obvious ifc= 0. Ifc >0, thencx Mif and only ifx M/c, which shows thatMis an upper bound ofcAif and only ifM/cis anupper bound ofA, so supcA=csupA.

6 Ifc <0, then thencx Mif and only ifx M/c, soMis an upper bound ofcAif and only ifM/cis a lower bound ofA,so supcA=cinfA. The remaining results follow similarly. Making a set smaller decreases its supremum and increases its thatA,Bare subsets ofRsuch thatA B. If supAand supBexist, then supA supB, and if infA, infBexist, then infA supBis an upper bound ofBandA B, it follows that supBisan upper bound ofA, so supA supB. The proof for the infimum is similar, orapply the result for the supremum to A B. Proposition thatA,Bare nonempty sets of real numbers such thatx yfor allx Aandy B. Then supA B. Sincex yfor allx A, it follows thatyis an upper boundofA, soy supA.

7 Hence, supAis a lower bound ofB, so supA infB. IfA, B Rare nonempty, we defineA+B={z:z=x+yfor somex A,y B},A B={z:z=x yfor somex A,y B}Proposition ,Bare nonempty sets, thensup(A+B) = supA+ supB,inf(A+B) = infA+ infB,sup(A B) = supA infB,inf(A B) = infA setA+Bis bounded from above if and only ifAandBare boundedfrom above, so sup(A+B) exists if and only if both supAand supBexist. In thatcase, ifx Aandy B, thenx+y supA+ supB,so supA+ supBis an upper bound ofA+Band thereforesup(A+B) supA+ get the inequality in the opposite direction, suppose that >0. Then thereexistsx Aandy Bsuch thatx >supA 2,y >supB follows thatx+y >supA+ supB 602. The supremum and infimumfor every >0, which implies that sup(A+B) supA+supB.

8 Thus, sup(A+B) =supA+ follows from this result and Proposition thatsup(A B) = supA+ sup( B) = supA proof of the results for inf(A+B) and inf(A B) are similar, or apply theresults for the supremum to Aand B. FunctionsThe supremum and infimum of a function are the supremum and infimumof itsrange, and results about sets translate immediately to results about :A Ris a function, thensupAf= sup{f(x) :x A},infAf= inf{f(x) :x A}.A functionfis bounded from above onAif supAfis finite, bounded from belowonAif infAfis finite, and bounded onAif both are and operations on functions are defined pointwise as usual; forexample, iff, g:A R, thenf gmeans thatf(x) g(x) for everyx A, andf+g:A Ris defined by (f+g)(x) =f(x) +g(x).

9 Proposition thatf, g:A Randf g. Ifgis bounded fromabove thensupAf supAg,and iffis bounded from below, theninfAf gandgis bounded from above, then for everyx Af(x) g(x) ,fis bounded from above by supAg, so supAf supAg. Similarly,gisbounded from below by infAf, so infAg infAf. Note thatf gdoes not imply that supAf infAg; to get that conclusion,we need to know thatf(x) g(y) for allx, y Aand use Proposition , g: [0,1] Rbyf(x) = 2x,g(x) = 2x+ 1. Thenf < gandsup[0,1]f= 2,inf[0,1]f= 0,sup[0,1]g= 3,inf[0,1]g= , sup[0,1]f >inf[0,1] limits, the supremum and infimum do not preserve strict inequalities Functions61 Example : [0,1] Rbyf(x) ={xif 0 x <1,0 ifx= <1 on [0,1] but sup[0,1]f= , we consider the supremum and infimum of linear combinations offunc-tions.}

10 Scalar multiplication by a positive constant multiplies the inf or sup, whilemultiplication by a negative constant switches the inf and sup,Proposition thatf:A Ris a bounded function andc R. Ifc 0, thensupAcf=csupAf,infAcf= <0, thensupAcf=cinfAf,infAcf= Proposition to the set{cf(x) :x A}=c{f(x) :x A}. For sums of functions, we get an , g:A Rare bounded functions, thensupA(f+g) supAf+ supAg,infA(f+g) infAf+ (x) supAfandg(x) supAgfor evryx [a, b], we havef(x) +g(x) supAf+ ,f+gis bounded from above by supAf+ supAg, so supA(f+g) supAf+supAg. The proof for the infimum is analogous (or apply the result for the supre-mum to the functions f, g).