Abstract approved: Dr. class of fundamental rational sequences. In fact, it is also possible

Transcription

1 AN ABSTRACT OF THE THESIS OF NANCY MANG ZE HUANG for the MASTER OF SCIENCE (Name) (Degree) i MATHEMATICS preseted o April 26, 1968 (Major) (Date) Title: A CONSTRUCTION OF THE REAL NUMBERS USING NESTED CLOSED INTERV LS Abstract approved: Dr..11. Arol It is well kowhthat a real umber ca be defied as a equivalece class of fudametal ratioal sequeces. I fact, it is also possible to defie a real umber as a equivalece class of sequeces of ested closed ratioal itervals. This paper is devoted to the latter case.

2 A Costructio of the Real Numbers Usig Nested Closed Itervals by Nacy Magze Huag A THESIS submitted to Orego State Uiversity i partial fulfillmet of the requiremets for the degree of Master of Sciece Jue 1968

3 APPROVED: Professor of, athematics i charge of major Head of o Departmet p of Mathematics Dea of Graduate School Date thesis is preseted April 26, 1968 Typed by Clover Redfer for Nacy Mag ze Huag

4 ACKNOWLEDGMENT Although it is difficult to fid words to completely express my gratefuless to Dr. B.H. Arold for his fatherly care ad patiet help durig the past two years, I must at least try to let him kow how much I appreciate his efforts to guide ad cousel me i a maer which has helped me to fit ito this America society. Thak you so much, Dr. Arold. Sicerely, Nacy M. Huag

5 TABLE OF CONTENTS Chapter Page I. INTRODUCTION 1 II. THE EQUIVALENCE RELATION IN F 3 III. ARITHMETIC OPERATIONS AND ORDER IN R 7 IV. COMPLETENESS OF R 20 BIBLIOGRAPHY 29

6 A CONSTRUCTION OF THE REAL NUMBERS USING NESTED CLOSED INTERVALS I. INTRODUCTION Cohe ad Ehrlich [1] have give a developmet of the real um ber system from Peao's postulates. The step from the ratioal umbers to the real umbers is made by equivalece classes of Cauchy sequeces of ratioal umbers. I this paper, we assume the developmet of the ratioal umbers as give i [ 1 ] ad we use their otatio, the we defie a real umber as a equivalece class of sequeces of ested closed ratioal itervals ad prove that our developmet is equivalet to that of Cohe ad Ehrlich. I [1], a iteger is defied as a equivalece class of ordered pairs of atural umbers, ad a ratioal umber is defied as a equivalece class of ordered pairs of itegers. I our costructio of the real umbers we agai begi with a equivalece relatio. this paper, a equivalece relatio will be defied i the set of all sequeces of ested closed ratioal itervals. A real umber will be defied to be a equivalece class uder this equivalece relatio. By suitable additio, multiplicatio, ad order, the set R of all real umbers will be made ito a ordered field which will be a extesio of the ordered field Q of all ratioal umbers. The order i R will have o gaps i the sese of Defiitio 3. 9 of [ 1 ]. I

7 Equivaletly, every fudametal sequece of real umbers will have a limit i R, i. e. the coverse of Theorem of [1] (which is false i Q by Theorem of [1] ) will hold i R. 2

8 3 II. THE EQUIVALENCE RELATION IN F Before we defie our equivalece relatio i the ratioal um bers, we eed the cocept of a sequece of ested closed ratioal i tervals. Defiitio 1: We defie {[a ' b]) 1 as a sequece of ested closed ratioal itervals if (1) a ad b are ratioal umbers. (2) a < b for all i N. (3) [a +1, b +1] C [a, b] for all i N. (4) L(a b ) = 0, where L stads for Lim 00 We shall write {[a, b ] } for {[a, b ] }, ad use F =1 to deote the set of all sequeces of ested closed ratioal itervals. Actually, from the defiitio of {[a, Ia)), we ca prove that (a) ad (b ) are fudametal ratioal sequeces. Theorem 2: If { {a, b ] } E F, the (a) ad (b ) fudametal ratioal sequeces. are {fa, b ] } E F meas that L(a b ) = O. Hece, for ay give s > 0 i Q, there exists (E) i N such that Ia b i < c for all > (c) i N.

9 4 But [a, b 1C [a, b ] for all m > i N, m m i.e. a < a <b <b forallm> i N. m m Therefore l a a l < l b a m i< ad E Ib b m < Ib a l<e for all, m > (c) i N. Hece, (a) ad (b) are fudametal ratioal sequeces. We are ow ready to defie our equivalece relatio i F. Theorem 3: The relatio ` i F defied by {[a, b ] } {[c, d ] } ' L(a c ) = 0 is a equivalece relatio. i) {[a, b ] } {[a b, ] } sice L(a a ) = O. ii) {[a, b] } {[c, d] } implies {[c, d] } {[a, b] } sice L(a c ) = 0 implies L(c a ) = O. iii) {[a, b ] } {[c, d ] } ad {[c di)} {[e, f, ] } {[a, b ] } {[e, f ] } sice L(a c ) = 0 ad imply L(c e ) = 0 imply L(a e ) = O. Hece, is a equivalece relatio i F. Theorem 4: If {[a b ]}, c,, d ]} E F, the L(a c ) = 0 if ad oly if L(b d ) = O.

10 5 If L(a c ) = 0, for ay give e > 0 i Q, there exist l, 2, 3 i N such that I a c I < 3 for all > l i N, Ia b I < 3 for all > i N, 3 I c d I < 3 for all > 3 i N. Let (e) = max {1, 2, 3}, the ib d i < Ib + Ia ci + Icdi < e for all > (e) i N. Hece L(b d ) = O. Likewise, we ca prove the other implicatio. L(b) =a. Corollary 1: For Ha, io]) e F, L(a) = a if ad oly if L(a) = a i. e. L(a a) = O. But L(b a ) = O. The L(b a) = L(b a +a a) = L(b a ) + L(a a) = O. Therefore L(b) = a. Coclusio: If two sequeces of ested closed ratioal itervals are equivalet, the their left edpoits ad right edpoits form two

11 6 fudametal ratioal sequeces; furthermore, if either oe of them coverges, the both of them do so, ad they have the same limit. Defiitio 5: A real umber is a equivalece class of the set F uder the equivalece relatio. We use C to {[a b de ] } ' ote the equivalece class cotaiig the elemet {[a, bill ) E F. We deote the set of all real umbers by R ad use, r),... for real umbers.

12 7 III. ARITHMETIC OPERATIONS AND ORDER IN R Before we defie our additio ad multiplicatio i the real umbers R, we shall show that the expressios we shall use for sum ad product of = C {[a, b }] pedet of the choice of Ha, b }] E g ad ad = C { [ c, d} ] { [ c, do }] E l. are ide the Theorem 6: If {[a, b ] } {[a b, l ] } ad {[c, d ] } {[c,d l ]}, (1) (2) Ha bñ+dñ I {[a+c, b+d] } {[ac, bd] } { [añc, bñdñ] }. }. (1) By the defiitio of, we kow L(a al) = 0 ad L(c c' ) = O. But L(a +c (a' +c' )) = 0 i Q. Hece {[a +c, b +d ]} ^ {[a.' cl, bl +dl ]}. (2) Sice (a ), (al ), (c ), ad (c' ) are fudametal ratio al sequeces, by Theorem i [1], there exist a ad c' i Q such that I a I < a ad I cl I < c' for all i N; by Theorem i [ 1], (aal) ad (c c' ) are fudametal ratioal sequece i Q. Sice L(a al ) = 0 ad L(c c' ) = 0, there are, for each positive e i Q, 1(e) ad 2(e) i N such that

13 8 Ia a I < 2cß i Q for all > l(e) i N ad Zc' I c cl I < 2a i Q for all > 2(e) i N. Hece la cac I<_ I I I a c I+ la a I lc' I< a _ e 2a + 2c cl for all > max {1(e), 2(e)} i N. Therefore L(ac a' c' ) = 0 i Q ad {[a c, b d ] } {[a cl, b dl j}. Theorem 7: There are biary operatios f ad g o R t such that if { [a, b ] } E ad { [ c, d ] } E 71, the (1) f(,11 = C{[a +c, b +d } (2) g('rl ) = C{[a c, b d ] } The sets f = ( 1)' C{[a+c, b+d] }) I, {[a, b] }E t, {[c, d], TIER ad g ((t'11)' C{[a c,b d ]})I{[a,b]}E,{[c,d]}Er,YIER are subsets of (R x R) x

14 9 If (t, TO E R x R, the t = C{[a ' b ]}' 1 = C{[c, a]) for some {[a, b }, {[c, d ]} i F. Sice {[a +c, b +d ]} E F, the pair ((t,i), 0 E f where = C{[ a +c, b +d ]} ((t3 1)) E f, the = C{[a'+c',b'+d' ]} { [a, b] } ' { [a', b' ] } ad where { [ c, d] } { [ c, dñ] } By the previous Theorem 6, it follows that {[a +c, b +d] a } {[al +c, b' +di ] } ad 5 _ ÿ'. Thus f is a mappig of R x R ito R, ad hece a biary operatio o R. If (t,i ) E R x R, the {[a, b]) E t ad {[c, d ]} El for some {[a, b ] }, {[c, d ] } i F. Sice {[a c, b d ] } e F, hece the pair ((e,i), 0 E g, where = C{[a c b d ]} ' If ' ((t, i ),,t)eg, the E, ' = C where {[a'c',b'd']} {[at,b'] }E, {[c', d' ] } E 1. By the previous theorem agai, it follows that {[a c, b d ]} {[a' c', b' d' ] } ad =,'. Thus g is a mappig of R x R ito R, ad hece a biary operatio o R. We are ow ready to defie our additio ad multiplicatio i the real umbers R. r, If Defiitio 8: We call the biary operatios f ad g of

15 10 Theorem 7 additio ad multiplicatio i R, respectively, ad write " +R r)" ad ot ri" for f(, r)) ad g(, q ). As usual, we shall feel free to omit the subscript "R "..R Now that we have defied additio ad multiplicatio i the real umbers R, we ca prove that (R, +, ) is a field. Theorem 9: (R, +, ) is a field. i) Associative Laws: If {[a, ] } E e, {[c, d] } er), {[e, f] } e the (+r)) + =C{[a (ai +c, b +d ]} c{[e +, f ]} C{[(añ C{[a+(c+e),b+(d+f)] } = C{[a {[a, b ] + C{[c +e,d +f ]1= } + (rl+), ad (.rl). = C{[a c, b d ]} C{[e, f ]} C{[(a c )e, (b d )f ]}, C{[a(ce), b(df)]} C{[a, b]} = O1' ) C {[c e, d ]}

16 11 ii) Commutative Laws: If { [a, b] } e, {{c, d] } E rl, the g + 1 = C{[a, + C{[c, = C{[a+c, di)} = C {[c +a, d+b] _ C{[ } + C{[ b I) =1+ g, ad g.,1 _ C{[a c,b d ] = C{ [ c a } = 1 g iii) Distributive Laws: If {[a, b] } e g, {[c, d] } E 1, {[e, f ] } E g, the (g+rl) = C{[a +c, b +d ]}. C{[e,f ]} C{[(a +c )e,(b +d )f ]} C{[a e+c e, b f+d f]} = C{[aea, bf] + C{ [ce' df] } _ + TI g. iv) Idetity Elemets: We ote that C serves as OQ, OQ the additive idetity, {[ 1 ] } as the multiplicative ide OR' C 1 Q Q tity, 1R; where OQ ad 1Q are the additive ad multiplicative idetity of Q, respectively.

17 NO Additive Iverse Elemets: For ay C {La, b ] } R, {[a, b ] }E F implies {[b, a] } E F ad {[ab, ba]} {[OQ, OQ]}... # OR 12 i C{[a, b] }+ C{[b,a] } C{[a b' ba } C{[OQ, 0 Q]}: R Hece C {[ b of C{[a, b]}, a serves as the additive iverse ]} C {[a, b ]} vi) Multiplicative Iverse Elemets: For ay C {[a b ] }# OR' ' we kow L(a) O. Sice (a) does ot have limit zero i Q, # there exists a positive elemet el i Q such that for every i N, I ak I > el for some k > i N. Sice (a) is fuda metal ratioal sequece, there exists 1 i N such that e Ia a1l < 2 I ak I 1 for all, m > l i N. If, for k1 > l, > el' the lal = lak (ak a)i > I lak lak al > el el el 2 = T for all > 1 i N. Hece a 0 for all > 1 i N. By the same process we ca prove b # 0 for all > 2 i N. Let é = mi (el, e2), ñ = max (1, 2), the

18 13 Ia I > 2 ad Ib I > 2 for all > i N. Take at = 1 for <, at =bl for >ri; b' = 1 for <, b' = for >ñ. a But for every e > 0 i Q, there exist (e) ad '(e) i N such that _2 ee IbbmI < 4 for all, m > (e) i N ad The ad Ia ^am I _2 ee < 4 Ia' a' I I bbm m Ib I Ib b' b' I I= for all, m > '(e) i N. 2 e (< 4 e for, m> ma.x((e), ñ) I m Ila Ia! la m am ( m 22 e e 2 i<_4 e for, m > max(' (e), ñ). e e 22 Therefore (a') ad (b') are fudametal ratioal sequeces. Sice {[a, b ]} e F, for ay e > 0 i Q, there exists ñ(e) i N such that Ia b I < ee 4 2 for all > ñ(e) i N. The

19 . 14 Ia bi Ia b I Iallbl 2 ee < 4 e e e for all > max(ñ(e), 1) 2 2 Therefore L(a' b') = O. O the other had a < a < b < b implies at < ai''1 +1 < b' +l < {[at, b' }e F. We fid b' for all i N, hece eé Ia a' Ia Ia bi 4 eé = b li IbI e 2 Z 2 2 for all > max(ñ(e), ñ), This meas (a a') has limit 1, i. e. L(a a') = 1. Now we have proved: for ay C {[a b ]} # OR' there ' exist fudametal ratioal sequeces (a') ad (b') such that {[a', b' ]} E F ad L(a a') = 1Q. Hece C {[a' b' serves as the ]} ' multiplicative iverse, 1 C{[a, brill, of C {[a b]}. This completes the proof that (R, +, ) is a field. ' We shall ow defie a order relatio i R. First, we defie positive elemets of R as the equivalece classes cotaiig positive sequeces i F. Before we do that, we prove the followig theorem.

20 15 oly if (b ) Theorem 10: If {[a, b ]} is positive. E F, the (a) is positive if ad (a) is positive, i. e. for some positive e i Q, there exists (e) i N such that a > e for all > (e) i N. Sice {[a b]} e F, we kow, for ay e > 0 i Q, there exists '(e) i N such that e e < b a < for all > '(e) i N. 2 2 But the b = b a + a > 2 + e = 2 for all > max((e), '(e)) i N. Hece (b ) prove the other implicatio. is positive. By the same process, we ca Defiitio 11: {[a ' b ]} is positive i F if ad oly if, (a) is positive. Notatio: We let R+ = eri _ ( ER1 for some {[a, b ]} E, {[a, b ]} for some {[a, b ]} is positive i E, (a) is positive. 1 the all Next, we will prove that if {[a, b ]} {[a ' b', ]} are e positive i F. E is positive i F, Theorem 12: If {[a, b ]} a {[a', bi "ill ad {[a, b ]} positive, the {[a', b' ]} is positive. is

21 16 By defiitio, we kow {[a, b ]} is positive if ad oly if (a) is positive. (a) is positive meas for ay e > 0 i Q, there exists (e) i N such that a > e i Q for all > (e) i N. O the other had, {[a, b ]} {[a', b' ]} if ad oly if L(a a') = O. That meas for ay e > 0 i Q, there 2 2 exists '(e) i N such that e < a a' < e for all > '(e) i N. The a' = a' a +a > 2 +e =2> 0 for all > max {(e), '(e)} i N. Hece (a') is positive ad {[a', bñ]} is positive i F. Now we ca say Corollary 1: R+ = {te R I {[a, b ]} is positive for all {[a,b ]Et }. Theorem 13: R+ is a set of positive elemets for R. We shall show that (1) + E R+ for all t,r1 E R +. 1 (2) t r) E R+ for all t, e R+. 11 (3) For t E R, exaclty oe of the followig holds: t E R+, = 0, t E R+

22 {[a, b ]} If t, ri e R +, the t= C {[a ' b ] }' C {[c d ]} where ad {[c, d ]} 17 are positive i F. Therefore (a) ad (c) are positive sequeces i Q. By the exercise i [1], we kow (a +c ) ad (a c ) g = C{[a+c, e ad + +c, b +d ]} that (1) ad (2) are fulfilled. If + are positive i Q. Hece = E R, so C{[a c, bd ]} = C{[a' by the Corollary of b ] }' ' Theorem 12, t E R+ if a oly if {[a, b ]} is positive i F if ad oly if (a) is positive i Q. = OR C {[0 0 ]} if ad oly if L(a) = 0Q. t = C e {[_b R if ad oly if ( b) a ]} ' is positive if ad oly if (a) is positive. Hece, by Theorem 3.26 i [1], exactly oe of t e R +, g = OR, g e R+ must hold. t Thus (3) is fulfilled, ad R+ is a set of positive elemets for R. t q = = Q, By Theorem ad Defiitio 3. 5 i [1], we have the followig theorems. tio i R. Theorem 14: The set T = {(t, l) Irl ; e R +} is a order rela Notatio: We write <R lit u (iii >R " if (t,ri) e T. Us ually, we omit the subscript R. Theorem 15: (R, +,, < ) is a ordered field.

23 So far we have defied additio, multiplicatio ad a order re latio i the real umbers R, ad we have show that (R, +,, <) is 18 a ordered field. Now, we are goig to prove the ordered field R of real umbers which we made is a extesio of the ordered field Q of ratioal umbers. Theorem 16: The mappig E of Q ito R such that E(a) = C is a isomorphism of Q ito R preservig {[a, a]} additio, multiplicatio ad order. E is a oe to oe mappig of Q ito R. For C {[a, a]} = if ad oly if {[a, all {[b, b]} i F if ad C {[b, b]} oly if L(a b) = 0 if ad oly if a = b. If a, b E Q, the E(a+b) = C{[a+b, a+b]} C{[a, a]} + C{[b, b]} = E(a) + E(b), ad E(ab) = C{[ab, ab]} C{[a, b]} C{[a, b]} E(a) E(b). Thus, E preserves additio ad multiplicatio. Also, a < b i Q if ad oly if b a > 0 i Q, so that (b a) is a positive sequece i Q. O the other had, C {[a a]} < C {[b b]} it R if ad oly if C {[b b]} C {[a a]} > 0 i R, so that C {[ba, ball

24 is a positive elemet i R, i. e. (ba) is a positive sequece i Q. Thus a < b i Q if ad oly if C i R, {[a a]} < C {[b b]} ad E preserves order. 19 We have just see that the ordered field Q of ratioal um bers is isomorphic to a subfield of the real umbers R. As usual, we will idetify Q with its image i R ad use iterchageably the symbols a ad C a, a ]}.

25 20 IV. COMPLETENESS OF R I this paper, first we have defied real umbers as equivalece classes of sequeces of ested closed ratioal itervals; secod we have proved that the set R of all real umbers is a ordered field which is a extesio of the ordered field Q of ratioal umbers. Now we are goig to prove that R is complete, i.e. every Cauchy sequece i R is coverget. Before provig this completeess, we eed some prelimiary results. Theorem 17: For every real umber e > 0 i R, there is a ratioal umber e such that 0 < e < s i R. Suppose E = C{[a b ' i R. Sice 8 > 0 i R, by defiitio, we kow (a) is positive i Q. The, for some posi tive e' i Q, there exists (e') i N such that a > e' P i R for all > (e') i N. Therefore, a > 0 i R for all > (e') i N, ad a e' Hece that C 2 e' e' {[a 2, b ]} 2 < b e' 2 for all i N. is a positive elemet i R. It follows e' s = C > C _ = l {[a,b.]} {[ 2, 2 ]} 2 e> 0. Theorem 18: (a) is a fudametal ratioal sequece i Q

26 21 if ad oly if (a is a fudametal ratioal sequece i R. By the previous theorem, we kow, for ay real umber E > 0, there exists e > 0 i Q such that 0 < e < E. But for this e > 0, there exists (e) i N such that la a I m < e i Q for all, m > (e) i N. It follows la a m I < E for all, m > (e) i N. Therefore (a) is fudametal i R. O the other had, for ay e > 0 i QC R, there exists (e) i N such that I Iaam < e i Q wheever, m > (e) i N. Hece (a) is fudametal i Q. With the aid of two theorems above, we ca prove the ext. Theorem 1 9 : If {[a, b ]} E i R, the L(a ) = L(b ) _ e. g = The proof is by cotrapositio. Suppose L(a) # f, ad C {[a' b' ]} ' {[a, b]} i R; we shall prove that L(a a') # O. Sice is a sequece of ested closed ratioal itervals, by The orem 2, (a) ad (b ) are fudametal i Q. From Theorem 18, we kow (a) ad (b) are fudametal i R too. But sice L(a), the there exists E > O i R such that a I > E i R for all large i N. I other words

27 22 ac{[a', b' }I > c or I C{[abñ, aañ]}i > e> 0 for all large i N. From Theorem 17, we ca fid e > 0 i Q such that c > e > 0 for large i N. But by the defiitio of absolute value (see page 68 of [1] ), it follows that I C{[abñ, aañ]}i = C{[abñ, aal',1]} or C{[a b', a a' ]}I ^ C{[a b', a a' ]} If C{[a, a_a]}i C{[ab, a_añ]} > e> = 0 for large i N, the (a a') is positive, therefore L(a a' ) # 0 ad {[a, b ]}. `5. If C{[a b', a a' ]}I= C{[a b', a al)} C{[a'a,b a e> 0 for large i N, the (a' a ) is positive; therefore L(a' a ) 0 ad {[a, b ]} j # `5.

28 23 We ow get a coclusio: for ay {[a, b]} e F if L(a) # t, the {[a, b]} ; i. é. for ay {[a, b ] }e F, if {[a, brill E i i R, the L(a) = t. But from the Corollary of Theorem 4, we kow L(a) = L(b) =,. This completes the proof. t Now we ca say for ay {[a, b ]} e F, we are able to fid a real umber t such that {[a, b ]} E t ad L(a) = L(b ) = t. Sice {[a, b]} ca oly belog to oe,, therefore we have a stroger statemet that for ay {[a, b]} E F, there exists a uique real umber ; such that {[a, b ]} e t ad L(a) = L(b) = g. The followig statemets are cosequeces of Theorem 19. Corollary 1: If t e R, for ay E > 0 i R there exists a i Q such that I ' a I< E i R. If g = C {[a b ' ]} i R, from Theorem 19, we kow L(a) =,. The, for ay E > 0 i R there exists (e) i N such that I e a I < E wheever > (e) i N. I particular, if a = a(e), the I a1< E is as required. Corollary 2: If g < r i R, there exists a i Q such that,< a <i i R.

29 24 Sice R is a ordered field, by Theorem of [1], the there exists,e R such that t <,< ri i R. Let e = mi { T10 > 0, the for this r, e R ad e > 0 i R there exists a i Q such that Is ai e< a < e i R. Therefore Y l; < E t< e< i R, i. e. a < +e<11. Corollary 3: R is Archimedea. For 0 < t < ri i R, let a ad b be two ratioal umbers such that 0 < a < t < b < t + ri i R. Sice Q is Archimedea (see page 69 of [1]) ad the embeddig isomorphism preserves additio ad order, there is a iterger i N such that a > b. But t > a > b > q i R, therefore R is Archimedea. Coclusio: From Theorem 2, we oly kow that for every se quece {[a, b ]} of ested closed ratioal itervals, the left ed poits (a) ad the right edpoits (b) form fudametal se queces. But ow we kow more. That is, they are coverget, ad coverge to the real umber which the sequece {[a, b ]} repre sets. We also kow every real umber is the limit of at least oe fudametal ratioal sequece. Fially, we kow that for ay real

30 25 umber, we ca fid a ratioal umber as ear the real umber as desired; o matter how close together two real umbers are, there are ratioal umbers i betwee. We are ow ready to prove the completeess of R. Theorem 20: R is complete. Suppose (p) is Cauchy i R, where = C P {[ap, bp]} ' the, for ay k i N, there exists k i N such that k k< < +k p k is true for all p > k i N. But, o the other had, k =C {[a k, b k]} L(ak) = = L(b k), k ad k k a k< < b k for all i N. Therefore, for ay k i N, there exist (k) ad ' (k) i N such that

31 26 I a k k + < k for > (k) i N ad i a k I< k for > (k) i N, respectively; k i. e. k a(k) k < < a < for > (k) i N k ad k k k < b < b (k) < k + k for > B (k) i N. Hece, for all m i N, there exist p = max {k I k < m} i N, am = max {a(k) k I b = mi {b + I k k< k m } i Q, < m} i Q, such that a m 1 < tp m < <b p m <b m 1 for all m i N ad P _> i N. pm But the we have {[a m' b ]} m C {[a m l' b m ad 1 ]} for all m i N,

32 27 I bmam i m m < I ba (m)+ m (a(m) m ) 1 = I? +b m m ' (m) m a(m) r) I <++ = m m m m teds to 0 as m teds to ifiity. Hece {[a, b ]} is a m m ested closed ratioal sequece. For this {[a1, bri }, we kow m m there exists some i R such that L(am) = L(b =. The m) g, a 1< (m) m a m ^5 < < b m < b m 1 ' (m) + m for all m i N; am 1< a m < b < bm + 1 < (m) m m p m ' (m) m for all m i N ad p > i N. Hece m I p I = I pbs m m m m + a(m)+ a(m) m m +13 < m m 1 m mm =4. I Sice R is Archimedea, 4 < e for some m i N. There m fore I I < E for large p i N. Hece R is complete. We have just proved that the ordered field R which we have costructed by usig ested closed ratioal itervals is complete ad

33 Archimedea. Although the way we build up the system of real umbers from ratioal umbers is differet from that of Cohe ad Ehrlich, our ed results (i. e. the field of real umbers) are isomorphic by Theorem 5. 3 of [ l ]. 28

34 29 BIBLIOGRAPHY 1. Cohe, L. W. ad G. Ehrlich. The structure of the real umber system. Priceto, N. J., Va Nostrad, p.