0,1 is an accumulation

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1 Sectio Accumulatio Poits Sectio 5.4 Bolzao-Weierstrass ad Heie-Borel Theorems Purpose of Sectio: To itroduce the cocept of a accumulatio poit of a set, ad state ad prove two major theorems of real aalysis; the Bolzao- Weierstrass Theorem ad Heie-Borel Coverig Theorem. Both proofs are two of the most elegat i mathematics. Accumulatio Poits Every set of real umbers has associated with it a set of accumulatio limit poits, a cocept which allows for a precise aalysis of closeess; closeess of real umbers, closeess of poits i R, closeess of fuctios, closeess of operators. The accumulatio poits may be a subset of a give set, part of a give set or totally disjoit of the give set. Its defiig characteristic is that every accumulatio poit of a set is ear some poit of the set other tha itself. Defiitio: A umber a is a accumuatio poit (or limit poit) of a set A if ad oly if for ay δ > the there exists the δ -eighborhood of a cotais at least oe poit of A distict from a. I other words every eighborhood of a cotais poits of A differet from a. Keep i mid that a accumulatio poit of a set may or may ot belog to the set. A eighborhood of a poit (ay ope iterval cotaiig the poit) that does ot cotai the poit is called a deleted eighborhood of the poit. Thus, the, 2 1 is a deleted eighborhood of 1. set ( ) { } Margi Note: Ituitively, a accumulatio poit of a set (which may or may ot belog to the set) is a poit where o matter how little you wiggle away from the poit you itersect poits of the set. I other words, the set likes to suggle up to accumulatio poits. Example 1 (Accumulatio Poits) a) Every poit i the closed iterval [ ],1 is a accumulatio

2 Sectio Accumulatio Poits poit of the ope iterval(,1 ) sice every deleted eighborhood of a [,1] itersects some poit i (,1 ). b) Fiite sets have o accumulatio poits sice aroud every real umber (iside or outside the set) you ca fid a deleted eighborhood that does ot cotai elemets of the set. c) The set (,1) { 2} has accumulatio poits [,1 ]. The umber 2 is ot a accumulatio poit of the set sice there exists a deleted eighborhood aroud 2 that does ot itersect members of the set. d) The set { 1/ : = 1, 2,... } has oe accumulatio poit at. Aroud ay other poit i the set you ca fid a deleted eighborhood that does t itersect the set.. e) The itegers Z have o accumulatio poit eve though the set is ifiite. This is easy eough to see sice each iteger is cotaied i a deleted eighborhood of radius.25 that does ot itersect ay members of the set. Margi Note: The reader may recall limits of sequeces from calculus which are examples of accumulatio poits of the elemets i the sequece. What do the Real Numbers Really Look Like? What does the real lie look like if you look at it really close up? We thik of it as a cotiuum of poits extedig idefiitely i two directios, but what if you could look at it uder a microscope ad were able to tur up the magificatio higher ad higher. What would you begi to see? You might be disappoited sice you will ever get to a stage where you would see, oe ratioal umber, three irratioal umbers, oe ratioal umber, The real umbers are self-similar, they look alike o matter what the scale. So how do we visualize the real umbers i our mids? Well, we simply have to uderstad the may properties of the real umbers which ca be verified mathematically, the use your imagiatio to visualize them it i your mid s eye. We ow come to oe of the most importat theorems i aalysis, the Bolzao-Weierstrass theorem, but before we state ad prove the theorem we must itroduce ourselves to the cocept of ested closed itervals.

3 Sectio Accumulatio Poits Nested ed Itervals By a sequece of ested itervals I [ a, b ] = we mea a sequece of closed itervals with the left edpoit a movig towards the right, ad the right edpoit b movig towards the left. The questio we ask is, what ca be said about the itersectio of all the sets; i.e. the set of all poits commo to every iterval? The followig lemma, which will be used to prove both the Bolzao-Weierstrass ad Heie-Borel theorems, gives the aswer. Lemma 1 (Nested Iterval Lemma) If a, b a, b... a, b [ ] [ ] [ ] is a ested sequece of closed itervals whose legths coverge to, i.e. lim b a =, the their itersectio cosists of a sigle poit ( ) [ a, b ] = x. = 1 Bolzao-Weierstrass Theorem Here is a iterestig questio that will test your ituitio about the real umber system ad accumulatio poits. Some people will aswer this questio i the affirmative ad others i the egative, so the questio is ot trivial. Here is the questio. Suppose you begi markig off poits iside some bouded iterval, ope, closed, or either, lets say [,1 ] for coveiece, ad suppose you do this idefiitely. The questio is ca you do it i such a way that there will ever be a accumulatio poit? I other ca you mark off poits i such a way that they ever buch up aywhere? Of course it is possible to mark off poits so you do have a accumulatio poit, simply pick x = 1/, = 1, 2,... which has a accumulatio poit at. I fact if x = you are clever, you ca pick a sequece { } 1 that has 2 accumulatio poits, i fact 3, 4,, a fiite umber of accumulatio poits. It is also easy to see that if the iterval is ubouded, say [, ) the there eed ot be a accumulatio poit, as the example x =, = 1, 2,... illustrates. So ca you

4 Sectio Accumulatio Poits x = fid a sequece { } 1 poit? Thik hard. of umbers i [,1 ] that does ot have a accumulatio Historical Note: Berard Bolzao ( ) was a Czech philosopher/mathematicia ad theologia. He was a Catholic priest who is best remembered today for his views i the methodology ad mathematics ad logic. May of the ideas later developed by Cator were uderstood by Bolzao. The Bolzao-Weierstrass theorem was first prove by Bolzao but ufortuately the result was lost. It was re-prove by the great Germa mathematicia Karl Weierstrass ( ). Weierstrass is ofte called the father of moder aalysis, havig brought mathematical rigor to the level we see today. The aswer to the questio about the existece of a accumulatio poit of a bouded ifiite set of real umbers is the statemet of the Bolzao- Weierstrass theorem. The theorem is importat ad the proof igeious. Theorem 1: Bolzao-Weierstrass Theorem Every bouded ifiite set S of real umbers has a accumulatio poit ir. Proof:, S a, b, Sice S is bouded, there is a closed iterval [ a b ] such that [ ] where the midpoit x = ( a + b) / 2 of [ a, b ] divides [, ] a b ito two closed subitervals. The subitervals overlap at the midpoit but all that matters is a, b. Now oe of these subitervals that their legths are half the legth of [ ] (or possibly both) cotais a ifiite umber of poits, else the set S is the uio of two fiite sets, cotrary to the assumptio that S is ifiite. Lettig I be the subiterval that cotais a ifiite umber of poits (if both 1 subitervals cotai a ifiite umber of poits we pick oe at radom), we cotiue by dividig I 1 ito two closed subitervals of equal legth, were we call I 2 a subiterval that has cotais a ifiite umber of poits. See

5 Sectio Accumulatio Poits Figure 1. itervals Cotiuig i this maer, we arrive at a sequece of closed [ ] a b I I I I, each of whose legth is half that of the previous iterval. Hece by Lemma 1 the set Ik cosists of a sigle poit, say k = 1 x. Strategy for the Bolzao-Weierstrass Theorem; Divide ad Coquer Figure 1 We are ot yet doe; we must show that x is a accumulatio poit of S. To show this let ( α, β ) by ay eighborhood of x ( α, β ), ad let k mi ( x α, β x ) is less tha k, we observe I ( α, β ). But I cotais a ifiite umber of poits ad hece so does the arbitrary eighborhood ( α, β ). Thus x is a = ad ote k >. Selectig a iterval I whose legth accumulatio poit of S. Margi Note: I terms of sequeces, the Bolzao-Weierstrass theorem says that ay bouded sequece x 1, x 2,... has at least oe coverget subsequece. Margi Note: Cator claimed that the Bolzao-Weierstrass is the basis for most importat results i aalysis. Realize the theorem is false if oe restricts oeself to ifiite bouded subsets of the ratioal umbers. The Bolzao- Weierstrass theorem states somethig iheret about the real umber system. Example 2 (Accumulatio Poits) a) The set A = 1,,,, is a bouded ifiite set so the Bolzao Weierstrass theorem guaratees at least oe accumulatio poit, which i this

6 Sectio Accumulatio Poits case there is exactly oe accumulatio poit, amely. The accumulatio poit of this set does ot belog to the set. b) set of itegers Z is a ifiite set but is ot bouded ad so the coditios of the Bolzao-Weierstrass theorem are ot satisfied, hece there is o guaratee of ay accumulatio poits. I this case the set has o accumulatio poits. c) The set A = (,1) { 2,3,4,... } is a ifiite set but ot bouded so the coditios of the Bolzao-Weierstrass theorem are ot satisfied; hece there is o guaratee the set has ay accumulatio poits. However the set does have accumulatio poits, amely all poits i the closed iterval [,1 ]. d) The set A = { 1, 2,3,4,5} is bouded but ot ifiite ad thus the coditios of the Bolzao-Weierstrass theorem are ot satisfies, hece there is o guaratee the set has ay accumulatio poits. I this case the set does ot have ay accumulatio poits. Fiite sets ever have accumulatio poits. Fiite Ope Covers C for Sets There are cocepts i mathematics which make you ask what does this have to do with aythig, but after further study you say wow, who ever thought of this! Oe such cocept is the idea of coverigs of sets ad i particular fiite ope coverigs. We will see that sets that have fiite ope coverigs behave to a great degree like fiite sets, ad of course aythig fiite is a lot simpler tha somethig ifiite.

7 Sectio Accumulatio Poits Defiitio Let A be a set of real umbers. A collectio C = { J : J } C of subsets of R is called a cover (or coverig rig) of A if A is a subset of the uio of the members of C, i.e. A { J : J C }. If each elemet J C i the coverig is a ope set, the coverig is called a ope cover, ad if C cotais oly a fiite umber of sets, the coverig C is called a fiite ope cover. You might thik of a coverig ituitively as a collectio of umbrellas providig shade from a summer su as draw below. Example 3 1 a) The family C =,1, : N is a ope cover for (,1 ), but o fiite sub-collectio of these itervals will cover (,1). Do you agree? b) The closed bouded iterval [,1 ] has a ope cover 1 1,1 +, : N the member of C whe = 1. { } C=, a example beig the sigle set ( 1,2 ) which is {,, : N} c) The real umbers R is covered by the ope cover C= ( ) but there is o fiite subcover forr.

8 Sectio Accumulatio Poits So why is the cocept of fiite coverigs so importat i aalysis? It has to do with the fact that sets which ca be covered with a fiite collectio of ope sets behave to a extet like fiite sets, which of course are easier to study tha ifiite sets. Compactess ad the Heie-Borel Theorem The cocept of ope covers of sets, i particular ope covers that have fiite subcovers, leads directly to oe of the most importat cocepts i aalysis, compactess. Defitio: A set A of real umbers is called compact if wheever it is cotaied i a uio of a family C of ope sets, it is also cotaied i the uio of a fiite umber of the sets i C. So ow the questio remais, how do we kow if a set is compact; i.e. every ope cover has a fiite subcover? As we have see i Example 3 some sets have this desirable property, some do ot. Offhad it would seem to be a very hard property to determie. Fortuately two mathematicias, Eduard Heie ad Emile Borel, foud a simple characterizatio of these fiite-like sets, whereby oe ca tell at a glace if they have this fiite property. This characterizatio is called the Heie-Borel (Coverig) Theorem. Theorem 2 (Heie ( Heie-Borel Coverig Theorem) a, b is compact. Proof: Every closed ad bouded iterval [ ] We must prove that for ay closed ad bouded iterval [ a, b ] that is covered by a collectio { J : J } subcollectio of itervals i C, say 1, 2,..., is for every x [ a, b] C= C of ope itervals, the there is a fiite J J J which also covers [, ] a b ; that we have x J k for some 1 k. The proof is by cotradictio. Assume the theorem false; that is, there, C= J : J C for which exists a ope cover of [ a b ] cosistig of itervals { } there is o fiite subcover J1, J 2,..., J. This beig true, the the midpoit of [ a, b ] divides the iterval ito two closed itervals, where at least oe iterval, which we call I 1, is ot covered by a fiite sub-collectio of members of the coverigc. We the divide I 2 i a similar maer ad arrive at a ew closed subiterval I 2, whose legth is half that of I 2 ad also is ot C= J : J C. covered by a fiite umber of members of the coverig { } Cotiuig i this maer, we arrive at a oicreasig sequece of closed itervals

9 Sectio Accumulatio Poits [ ] 1 2 a b I I I, where each iterval I k is half as log as its predecessor ad is ot covered by a fiite umber of members of the coverig { J : J } 1 we kow that the itersectio sice x [ a, b] ope itervals, say ( α, β ) such that x ( α β ) C= C. But from Theorem Ik cosists of a sigle poit, say k= 1 x. Now we kow there exists (at least) oe member of the family C of. But from the way the, itervals I k are formed there exists a iterval I whose legth is so small that x I ( α β ) { J J }, C= : C. A cotradictio; we kow I caot be covered by a fiite umber of members of C, but it is covered by ( α, β ), a sigle elemet of C. Figure 1 But this is a cotradictio sice we have said that I caot be covered by a J : J I α, β C= J : J C. fiite umber of coverigs of C= { C }, but ( ) { } Hece, we coclude every ope cover of a closed ad bouded iterval [ a, b ] does have a fiite subcover. Margi Note: The observatio that subsets of real umbers have fiite covers is equivalet to beig closed ad bouded was first observed by Germa mathematicia Heirich Eduard Heie i the 187s ad later i 1894 formulated precisely by Frech mathematicia Emile Borel.

10 Sectio Accumulatio Poits Note o the Heie-Borel Theorem ad Compact C Sets The Heie-Borel theorem as stated i Theorem 2 is a special case of a more geeral Heie-Borel theorem 1 which states; A subset of R is compact iff it is closed ad bouded. We stated a specialized versio of the theorem a, b ad we oly stated the theorem oe way; for closed itervals [ ] [ a, b] compactess whe i fact the theorem goes both ways. The Heie- Borel is importat sice it completely characterizes compact sets; i.e. closed ad bouded sets. Relatioship Betwee the Bolzao-Weierstrass ad Heie Borel Theorems The Bolzao-Weierstrass ad Heie-Borel Theorems are more closely related tha oe may thik. I fact if the accumulatio poit(s) of the ifiite set A, which is guarateed by the Bolzao-Weierstrass theorem, belogs to A, the the set is closed ad bouded, ad thus by Heie-Borel have fiite ope covers. The followig theorem, which will left uproved 2, makes this relatioship precise. Theorem 3 If A is a set of real umbers, the the followig are equivalet: i) The accumulatio poit(s) of A belog to A. ii) A is closed ad bouded. iii) A is compact. 1 We stated that closed ad bouded itervals have fiite ope subcovers whereas i fact all closed ad bouded sets have fiite subcovers. Also the coverse if true; if every ope cover of a set of real umbers has a fiite subcover, the the set is closed ad bouded. 2 The proof of this theorem ca be foud i most textbooks o real aalysis. A good textbook i this geera is Real Aalysis by Frak Morga, America Mathematical Society (25).

11 Sectio Accumulatio Poits Problems 1. Fid the accumulatio poits of the followig sets (if ay). State whether the coditios of the Bolzao-Weierstrass theorem hold. a) N b) Q c) R 2, 4 4,5 d) ( ) ( ) { 1 : N } e) ( ) f) Q g) (,1) m h) : m, N 2 1 i) m + : m, N 2. (Covers of Sets) What does it mea for a family of sets ot to be a cover C for a set A? What does it mea for a cover C of a set A ot to have a fiite sub-cover. Give examples of each. 3. (Compactess) Use the geeral Heie-Borel theorem, which states that a set of real umbers is compact if ad oly if it is closed ad bouded, to determie which of the followig sets are compact. a) { 1,2,3, 4,5 } b) [,1] [ 2,3] c) { x : x 2 = 2} d) [,1 ) e) [,1] { 2,3, 4,5} 4. (Closed Sets) A set is closed if it cotais its accumulatio poits. Fid the accumulatio poits of the followig sets ad verify that those sets that are closed do cotai their accumulatio poits. a) N b) Q c) R 2, 4 4,5 d) ( ) ( )

12 Sectio 5.4 { 1 : N } e) ( ) f) Q g) (,1) m h) : m, N 2 1 i) m + : m, N 12 Accumulatio Poits 5. (Ope Subcover) (Ope Subcover) Fid a fiite ope subcover of the set [ ],1 for the cover C =,1,. j 1 4 j= 1 6. (Itersectios of Closed Itervals) The itersectio of a fiite umber of closed itervals is oe of three types of sets. What are they? 7. (Itersectios of Ope Itervals) The itersectio of a fiite umber of ope itervals is oe of two types of sets. What are they? 8. (Examples) Give examples of the followig. a) A bouded set with o accumulatio poits. b) A ubouded set with oe accumulatio poit. c) A set with two accumulatio poits. d) A ubouded set with a ifiite umber of accumulatio poits. e) A ubouded with oe accumulatio poit. f) A ope set with o accumulatio poits.

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