Ambar N. Sengupta March, 2014 - LSU Mathematics

1 Notes in Introductory real AnalysisAmbar N. SenguptaMarch, 20142 Ambar N. SenguptaContentsIntroductory Remarks ..51 Ordered Fields and The real Number Fields .. Relations .. Fields .. absolute value function .. Archimedean Property .. real Number SystemR.. Maximality and the Completeness Property .. ofRand measurement .. 20 Problem Set 1 .. 202 The Extended real Line and Its extended real line .. of points for a set .. , Exterior, and Boundary of a Set .. Sets and Topology .. Sets .. Sets and Closed Sets .. sets inRand inR .. of a set .. The closure of a set is closed .. the smallest closed set containingS.

2 Is compact .. Compactness of closed subsets ofR .. The Heine-Borel Theorem: Compact subsets ofR.. 3634 Ambar N. Sequences .. Limits points of a sequence .. Bolzano-Weierstrass Theorem .. Limit points and suprema and infima .. Limit of a sequence .. Simple Examples of limits .. The sequence 1/n.. The sequenceRn.. Monotone Sequences .. The limit of a sequence is unique .. Convergent sequences and Cauchy sequences .. Every Cauchy sequence is bounded .. Every Cauchy sequence is convergent .. The rationals are countable .. The real numbers are uncountable .. Connected Sets .. 553 the Riemann Integral.

3 Sums .. of the Riemann Integral .. Partitions .. Darboux Criterion .. functions are bounded .. of a Function .. algebraR[a,b].. [a,b] R[a,b].. The Integral as a Non-negative Linear Functional .. Additivity of the Integral .. Monotone Functions are Riemann Integrable .. Riemann Sums and the Riemann Integral .. 83 Bibliography .. 86A Question Bank89 Notes in Introductory real Analysis5 Introductory RemarksThese notes were written for an introductory real analysis class, Math 4031, atLSU in the Fall of 2006. In addition to these notes, a set of notes by Professor were are several different ideologies that would guide the presentation ofconcepts and proofs in any course in real analysis :(i) the historical way(ii) the most natural way(iii) the most efficient way(iv) a comprehensive way, explaining the insights from several different ap-proachesThe reality of constraints of time makes (iii) the most convenient approach,and perhaps the best example of this approach is Rudin s Principles of Mathemat-ical analysis [5].

4 The downside is that there is little possibility of conveying anyinsights or studying the notion of completeness, a choice has to be made whether totreat the Cauchy sequence point of view or the existence of suprema as funda-mental. I have chosen the latter; it conforms to the classical geometric notion ofa positive real number being specified by quantities greater than it and those lessthan it. This point of view also guides the choice of approach in the treatment ofthe Riemann integral; the Riemann integral of a function is the unique real numberlying between the upper Riemann sums and lower Riemann notes here do not include a chapter on continuous functions, for which wefollowed the Richardson notes have not been proof read carefully.

5 I will update them time totime. Comments from many students have helped improve the notes. Amongthose who deserve thanks are John Tate (in-class comments) and Daniel Donovan(email 2014).6 Ambar N. SenguptaChapter 1 Ordered Fields and The RealNumber SystemIn this chapter we go over the essential, foundational, facts about the real real numbers arose from geometry in Greek Mathematics , as ratios ofmagnitudes, such as segments or planar regions or even angles. In the discussionbelow we focus on Euclid sElements, a segment EF is taken to exceed a segment GH, symbol-icallyEF>GHif EF is congruent to a segment GK, where K is some point between E and F. Animportant feature of this order relation is encapsulated in thearchimedean axiom:given any two segments, some multiple of any one of them exceeds the aAa pair PQ and RS if for any positive numbers n and m, the segmentnAB (which is n copies of AB laid contiguously) exceeds the segment mCD ifand only if the segment nPQ exceeds the segment mRS.

6 The ratioABCDis then essentially the equivalence class of all segment pairs which are in the sameratio as AB is to CD. Euclid also defines the ratio XY/ZW to begreaterthan theratio PQ/RS :XYZW>PQRSif they are unequal and if whenever mZW>nXY then also mRS> N. SenguptaA special case is that ofcommensuratesegments: if a whole multiple of AB,say nAB, is congruent to a whole multiple of CD, say mCD, then the ratioABCD isrational, and is denoted is readily checked thatmn=pqif and only ifqm= ratios are therational numbers. Other ratios areirrational. In either case,Euclid s considerations suggest that a ratio of segments may be understood interms of a set of rational numbers, for example the set of all those rationals whichexceed the given axioms of geometry, and the Euclidean construction procedures, showthat ratios of segments can be added and multiplied and, when 0 and negativesare included, an algebraic structure called afieldemerges (this is discussed atlength by Hilbert [3]).

7 The maximal such field, respecting the axioms of geometrypertaining to the order relation and congruence, constitutes thereal number aside the historical background, the real number system may be con-structed by starting with the empty set, constructing the natural numbers, then therationals, and then the real numbers by Dedekind s method of identifying a realnumber with a splitting of the rationals into two disjoint classes with members ofone class exceeding those of the s method has beed generalized in a striking, and vastly more power-ful way, by Conway [1], who shows how the Dedekind cut method can be appliedto abstract sets leading to the construction of all real numbers as well as tran-scendentals and infinitesimals.

8 Knuth s novel [4] is an unusual and entertainingpresentation of this Ordered FieldsIn this section we define and prove simple properties of fields, ordered fields, andabsolute values. The reader wishing to move on to properties of the real numbersmay skim the contents of the first few subsections, and proceed to in Introductory real FieldsAfieldFis a set along with two binary operationsAddition :F F F:(a,b)7 a+b( )andMultiplication :F F F:(a,b)7 ab( )satisfying the following axioms:1. The associative law holds for additiona+(b+c) = (a+b)+cfor alla,b,c F( )2. There is an element 0 F, thezerooradditive identityelement, for whicha+0=a=0+afor alla F( )3. Every elementa Fhas anadditive inverse a, called thenegativeofa:a+( a) =0= ( a)+a( )4.

9 The commutative law holds for addition:a+b=b+afor alla,b F( )5. The associative law holds for multiplication6. The associative law holds for additiona(bc) = (ab)cfor alla,b,c F( )7. There is an element 1 F, theunitormultiplicative identityelement, forwhicha1=a=1afor alla F( )8. Every non-zero elementa Fhas anmultiplicative inverse a 1, called thereciprocalofa:aa 1=1=a 1afor all non-zeroa F( )10 Ambar N. Sengupta9. The commutative law holds for multiplication:ab=bafor alla,b F( )10. Thedistributive lawholds:a(b+c) =ab+ac,(b+c)a=ba+cafor alla,b,c F( )11. The element 1 is not equal to the element 0:16=0We have not attempted to provide a minimal axiom set, and some of the axiomsmay be deduced from others. For instance, the commutativity of addition can bededuced from the other of the associative laws, we will just writea+b+cinstead ofa+(b+c), andabcinstead us note a few simple consequences:Theorem 1If x Fis such thata+x=a for some a Fand y Fis such thatby=b for some non-zero b Fthenx=0andy= particular, the additive identity and the multiplicative identity are unique.

10 More-over, 0=0and1 1=1 Notes in Introductory real Analysis11 Proof. Adding atoa+x=ashows thatxis 0. Multiplyingby=bbyb 1shows thatyis 1. The other claims follow from0+0=0and1 1= 2If a,b F, and b6=0, then ( a) =a,and(b 1) 1= ,( a)b= ab,and( a)( b) = multiplicative inversea 1is best written as the reciprocal:1b=b 1,and the productab 1asab=ab 1 There are many other easy consequences of the axioms which we will usewithout denote the set ofnatural numbersbyP:P={1,2,3,..},( )and the set ofintegersbyZ={0,1, 1,2, 2,3, 3,..},( )We can multiply any elementa Fby an integer as follows. First define1a=a,where now 1 is the number one inZ. Next,2adef=a+a,12 Ambar N. Senguptaand, inductively,(n+1)adef=na+a( )for alla Fand alln P.