~eI) - personal.maths.surrey.ac.ukpersonal.maths.surrey.ac.uk/S.Zelik/teach/completness.pdf ·...

9
l?eeC<..:b-e.. ) ~ EJnJr,c{ E:Jc: -- ,) f ( r>,' ,r-,Yv-GD &6~, . L e.d1A-~ 1-: Corn/?- L-0Ln~ CLn G ~"-'" '/_ - 0 .. ( (.':1 ) r ))' (JR I ;. 7)) ([ CL/'~/J;J, j, !) , ( (Q, ~)/ j. /) LXcuYLp~. q , t-- r ._ GtP~ m et ?-/~ ·spa U2Y:>. J n )0 C1JL if' C U./~C'AJ7) )( '= (Q, (5) w; I){ '!,-loJI,,"~cJ meLUL wf?d i -s £4 X = [Cl, is 1 cJ.,l.f=J2rJc12- &Lw-een )( OJ? cl X.P If we c-,o/_oI€1>~ '>( C IR OJId XCII<.. tflfC'-n. X ; -'5 ~ CV)~ X I '" c. ~"d - tfk df,'~--V7CI2_ /':5 c.£zo.rc. c Pxd I1.t u.Y:o, c..GlY?fclerc. (X, I f) ~ an.c1 eX", I I) cu:" /mnf: I c) spa.=. Tf?eY2- ~,x , Qnc{)( cv e- BOTI-( ~,P'U7 and e&".l 9 1;;0 cl; ff~n ce ?" But ev~ r::.cmLnl4~ [2UJJcilorc f X -0, 11':< 13 &'6U7e::kd on of CA cA; eN eY, i L -3 m i,) IIn "t"'y, D.noL n1 cvy ; It i W'Y} . --. ~'- I f,ai /3 m-t -L -ut e. f"",,- me 'S f'0< C'-Q... X I .? ITfieJ2.Q.. i"" CV) iniL(I,,'h~ cJi;f/~=- r;,dw~ X Cuw{ Xl,' /-{ovv -to J~fk_ 11;;>; cA,P-fb0:2nee.? GJ - i"", ,. go..cI. " f<or><.. iAe_ csJ!c.tJ."Q, pcxnt 01- V/evv 0YJ.dl IR 'so, "8 ood "- INr; 0~ , 'S {h"'- cAf.f~ eJ2. ?' ~ lO. __ . ~ ,=rYi.(O~ '?~ P?O~ ~ ( !-I x ICJnc C. fen... tR) ~ ~eI) x,,- E JR ,. 'S c: Cctu~ C&Jo (X, oil i'S a rndue- $F~Ce "'3'€~~ 1(7.. 11<.. i ff!xn EX, I"" c:- ,Chu1d '3£rpuQl7=_ n I pp V [>0 3. N =j/{f) ~cVct-fJcet V t. > 0 :l N == 1\/ ( f..) SMcYc. , ~ _ {R ed.- -------- - i 0( (X TL.) Xm-rn ) < 2 /1 X re. - )( n+h< I < 2.. , I 1'1>/ /V( 2.) m G N rL ~ /\/CE-) mej\T! ._ _) __ ) I R efo"-i1? /,( Ca,L 00. X' r~ I 0, CL Ca. lA ~ '5 e '1;"en "-Q.. in. X I pp VE>o 3~EX an..cf. IV== N(E) 3uJL -tJ?a± .._--- - tX/1.:' 8r'(X,,- ') if ~ >NU:) I e. ¥e cJ:: -LA e. e:y /AI v~ CJ<.. 0S0, GIn E- X. i Xn -'>Xo In... X ,pt V€ >0 -=J N= ;V(~) j--x f) - ') ,'() n ') AI(~) I ' rL E- 1~£.'(Xo J ,

Transcript of ~eI) - personal.maths.surrey.ac.ukpersonal.maths.surrey.ac.uk/S.Zelik/teach/completness.pdf ·...

Page 1: ~eI) - personal.maths.surrey.ac.ukpersonal.maths.surrey.ac.uk/S.Zelik/teach/completness.pdf · l?eeC,'~"-'",r-,Yv-GD'/_

l?eeC<..:b-e.. )

~ EJnJr,c{ E:Jc:

-- ,) f ( r>,' ,r-,Yv-GD &6~, .L e.d1A-~ 1-: Corn/?- L-0Ln~ CLn G ~"-'" '/_

- 0 .. ( (.':1 ) I· r ))' (JR I ;. 7)) ([ CL/'~/J;J, j, !) , ( (Q, ~)/ j. /)LXcuYLp~. q ,t-- r ._

GtP~ met?-/~ ·spa U2Y:>. J n )0 C1JLif' C U./~C'AJ7)

)( '= (Q, (5) w; I){ '!,-loJI,,"~cJ meLUL wf?d i -s £4X = [Cl, is 1 cJ.,l.f=J2rJc12- &Lw-een )( OJ? cl X.P

If we c-,o/_oI€1>~ '>( C IR OJId XCII<.. tflfC'-n. X ; -'5 ~ CV)~

X I '" c. ~"d - tfk df,'~--V7CI2_ /':5 c.£zo.rc. c

Pxd I1.t u.Y:o, c..GlY?fclerc. (X, If) ~an.c1 eX", I I) cu:"

/mnf: Ic) spa.=. Tf?eY2- ~,x , Qnc{)( cv e- BOTI-( ~,P'U7 and e&".l9 1;;0 cl; ff~n c e ?"

But ev~ r::.cmLnl4~ [2UJJcilorc f X -0, 11':< 13 &'6U7e::kdon of CAcA;eN eY, iL -3 m i,)IIn "t"'y, D.noL n1 cvy ; It iW'Y}

. --. ~'-I f,ai /3 m-t -L-ut e. f"",,- me 'Sf'0< C'-Q... X I

.? ITfieJ2.Q.. i"" CV) iniL(I,,'h~ cJi;f/~=- r;,dw~ X Cuw{ Xl,'/-{ovv -to J~fk_ 11;;>; cA,P-fb0:2nee.?

GJ - i"", ,. go..cI. " f<or><.. iAe_ csJ!c.tJ."Q, pcxnt 01- V/evv0YJ.dl IR 'so, "8ood "- INr;0~ , 'S {h"'-cAf.f~ eJ2. ?'

~ lO. __

. ~ ,=rYi.(O~ '?~ P?O~ ~ ( !-I x ICJnc C. fen... tR) ~~eI) x,,- E JR ,.'S c: Cctu~ C&Jo (X, oil i'S a rndue- $F~Ce

"'3'€~~ 1(7.. 11<.. i ff!xn EX, I"" c:- ,Chu1d '3£rpuQl7=_n Ipp V [>0 3.N =j/{f) ~cVct-fJcet

V t. > 0 :l N == 1\/ ( f..) SMcYc. , ~ _

{Red.- -------- - i 0( (X TL.) Xm-rn) < 2/1 X re. - )( n+h< I < 2.. , I 1'1>//V( 2.) m G N

rL ~ /\/CE-) mej\T! ._ _) __) I

Refo"-i1? /,(Ca,L 00. X'r~ I 0, CL Ca. lA ~ '5e '1;"en "-Q.. in. X I ppVE>o 3~EX an..cf. IV== N(E) 3uJL -tJ?a±.._--- -

tX/1.:' 8r'(X,,- ') if ~ >NU:) Ie.¥e cJ:: -LA e. e:y /AI v~ CJ<.. 0S0, GIn E- X. i

Xn -'>Xo In... X ,pt V€ >0 -=J N= ;V(~)j--x f) - ') ,'() n ') AI(~) I' rL E- 1~£.'(Xo J ,

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G- \..L' - ~J . 'Sk:' ·t. (.(..L, ( ( ~.(ncn...·- CCJn· cen-ttc fbil2,. J

R eJY}I'nJ~: X:::::. /R ReJ/-..12- w-e- gewe.

O.JYd·CfYG. , (A x /.om... e . ~ ~It 1 )I

,fo'L 0fI,5j _go_'un_o~_ec/_ '3ei e, cif< -tJ?e=.- E¥i.;L~ 0'f ~ (

-L~ef0,- I we r:1wve- It ~l~ _.~~

COJ-LeJ~L Tge<5WFf<.. : AnI,{ CQ..{AcQ.,(.1 '3fGOMNlCS2- I n fR. gc~ 0-. B/?v'·-b'l() '"'}(J y (c-'Omp-p£tn~)

; An)( r:'0/76-6oaL &unded SQ if"L'2/7CfL f!?os,Z CL (?jrn/t:

p~eiXJJ1O-\ ,,--,fJvo;tzo..v.:;, • A n~ ~deJ. S~<.:.SL i "-- ~ CJ:'JI?~ 'rJ S-I.JJ0Jl..QrY( C\., conv~t; s>u~~

( C.OTY1p~Jn~~ )

A et i~~ ~oi.e.fl7'</Jts ~'UL ~CjMiveJiur6. in- JR.. THe ?J2..~<Dn

JR i~ IIt.OO ym~il OJ?d.. eN~':3 I'~ ;'m1xedl/ ~'-Q.l!V<. 4L~/" -".....l./)~ . - <. L. /)j( '-I..- J',-.j) ,I. l In" i /'- '-/d , _11

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LetX=- c[ Cl I g 1 &- a. '~PQc:..f2-

-tund/an-3:, f: [0/ g1'~ /£2 w;-l:.k

1VI}~~;:= mcvx (-fex) r I, L XG[CI,@j

~ (ffp~ ~,--nC)~

~=rn: (CLa,eJ, /i 1~00) I"" <:;Q.~ > 1 eo I ~[Q, e)wit.rih-e ~-n~ '''0, Q BANACl-{ ""f''''= 0

r:z.a:! L e -t irL E. X ge Q CCUAc.% S',J2JtA/~ i'; >< w~neszd. -to con ~7M d CL &rn /i-fu...n C5U OrL-f !C3L .!J-zo:t: '9,.-e(~I[IlQJ1c.Q,

L e i Xo ~ [G I g :1 f5e... oscJ6; i~. Con «',./ deJL CL '3J2q;tAQ17.e.e~ 0(2nu_m ~ ~ .-f?n. ('X0). ~j) ce-.

fJ7rL( Xo ) -K,,," (Xc \ I~ mOJ>< J -Pr<- 0<) -fnNn (X) k Jlf>~fm I~""X , L

an~ f;:i '3 Cl- Cu,0J ~<:Q.. irL. C[o>G"1 ::y £-fn (XcJ5t)&A/ ig,

CL CCLU~ ~c.s2- ;'17- ,J!( ,f3.v:;6 fiR I ~ C-orYJ~

'5> \f Xo E [Q) g ~ :1 IF (Xo) -, '=:, ~rn fA- (Xo))n4\;)0

/Vee.cJ -/:::0 cJ'w.&: r) tf;;~ e~n~71't.; ~ iAJI/k'Ln, ,'\ ,D . I----- 2.. ) I I -S C\.. C """17 tnUlC'-C,- S

@; q.,,~ f" ,,,,"'- Ca-c\~ ~ =- "? Vc:>0 3 /V", A/( E-~U7 0&&/70

\lX E [Cl B ~ i--Pr c' X') Cl ( '\ I . '0 () . ' .I • i?.. -lOrrY( X) ~ /!--fn-jl/-tY"YrltD>O <El CJI&/l/n':>;,Jvcf)

, P~.9>i'?J ,io ·t~--e. elt?1I'-6- m4,OO I 'we. ~e.- L }

~jJ1/'~ c0Ylv.en~(}lf)1ci\ 1/ -Pr< (X) - f(X) I < ~ n >...!V(E) '\7 X GCq 6'\C ,/><;;, f'ICOIeCl <::J ') i I ....)

® A {U);fC5?YY( em;t of, cc-n-f.Jr:u~ -rlnJ;~'S /''3, et corlilnr/i..~

-f~;e;:tr( , P CrLQ..-~Q } cJ e-flz,' Q

,mcn..e.. f'xa,npf--e:- (~RD /\/GVV'S) ~ X ~=C [CAI g.1 wii g fk /n-Le;ra.1?-

nC5'ifn. "ltfI1! {":;:: f i fc><() I Jx - I ~ NoT c.om~e. /- -'

IndeeoC, "1ed. [a, @ ~ ~ [-I, IJ OJ1J ed {l (Po!ni-w,'-;e)fI~--.- .. ' 'Tf?erL In... -7 ex)::::. gan..X-i?(x) / CJ ..

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v/\/) ..

L erYUYLCL . (e 2.) 11· II~) I cs et. g,OJI.QQ f{ '!.PCAr:sz .

P2.=f Let X re. ~ ( X, ': X:, ...) 8e Cl- Co:.u..~ '3.€'f/'€/iCQ dl e2.

fIeeo( Lo FLC V'2 '3 X 0E ~ <uccA. -f1!ai:: lx'~ In_ ere..lA 5/~ J -iRe /f7e~uair' t(j

i' 'x ,i?..~ x' I/..:tne. } fL IL - L '::::. 11 'x- X fl+1I( lIe?.. L G A/

wuAIJ, at x '1 i'" 0.. Co..u.J/" ~ ~i7 Le.. In f''<.. ,vV e. "3, eQ.. t fat) n'~ cJ4,',d2. X i .':J I'lE,{/ ~ Cl- ~ :2:1 se.~ CQ_ ifL If;! ..~

\7 i e /\/ ; x/ ;""~~ ';(,"1- e.x Id~ (c=<d,,,a1e _WiK e/», t )---- :.J

/Veed -to fy2....0V~

(j) XO,,= ( 'X tJ . 'x' ,0) E e,I)" ) (, '--

® /1 X r:.... x 0 ({~ 0 rL-:::> r;;;o)

® 'SUi "-"- X n i '3, ~ ~~ _ '1£'jMJ2./IClZ.. in ~ o..n cA

/1 ~/I_=-IIX:~ 11 I .~ IJ X n_ X "tt" I IJ

.:> 11X" If ; 'S CL C'cw~ ~c.e.. in. 11<. '> 11X'<II 1'0 C.on vengd.'S> Jj X le. 11 ;::, CL 8ou.n ...deJ. ~eCjlI"l.Qn C-L',- ~

'> 'if:(x/) 2. ~ 11 X n "e'\- ~ ~L"=- I

n t I De.' ~m/t(-'C~~ '0 trJ It (Xn'- ~ cl'-- -- - -

'to tRe e,rrli't IV -7 00

;.;;' (;h2. ~ c: JL~J ,

~~

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®1\ x~ xn+rn.n~'L ~ g), reo, NI£}, ~!Bl . ., ntGA/

'.

P~3 t.o ~~efV/·.Lmt m..~ij:J

.L:~ (XiIL.- x/ )2. ~ ~<.':::..1

rv\ eN n >" /\I(€.)~ )

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J-n. CL -fad) A/VY nC9n..- c.o;rp(Q;te.. ~paCQ C0J7 fk~ (JJJ)

o g i:evned IrL cA ~ vv~ .

(X) ;;r) i· 53 ~ 'U)I>;;,~t/Ot~)~po.Cf2.- (X/ cl) /f

~

CLnJ.. X I~ COfl7~

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E x~. I) f~e- CD~h61L ofZ ex I "S IR

2) tfire-. C-OIn~/CY?- C)·f 6(fL / S1 rR i?..

~) Let X =: P [a,e] - spo,CQ of oL£ ~0fi7-3

(') 1'<.- [Cl) g) I -- ~'-. 17 .!f (x) = CL re X +- + Qo ) i IGG A/

(PCo,g] 11 '1Ii~) I~ not: c.om~E'-J h 'X f71 X I.' ().1- x, t () Je = e·/'YC L -. . L9-u/L e.. iSS no . Cl moUVl6-177.(14>00· <.. I / . 0

L-=D .

wRaf. I», the CDrY2f' eel/OIL of PLo., I?J ?

BJ iRe- Werercd2IV0, t~eCAAn() pea);] /s:,

.Je,n~~ /n. C [Q) BJ (eur~ Lont/nul~ .fCUld/OrLLOJl 6-e ogieuned.. ~ cc (AJ7/·fc.r-uY? 6-/?'7Ji o!j;)00no7YI53)

. 5> iJ5e- CJ5fn;:Jld.>/ofL of P[QJS] /~ e[a)?J(c·; ~I w/tfL ~d in ·-L/~e 9yur·-me-i?/~),

C0LOf~. THe 5fO-Cl:L ( C [Q, e]) It· IlL""') /-'3S' epW'LQ 'g~.

~. ; .LnJee.J.) tJ?e PO~OrYI'3, w,tFL zal.;0nd9 ce-eff/c/-eni~ OJLQ. Jen~ -e... i·1?- pro) g J wed t~~

~ _<'..OlU1t~ mCif/21 ~of '3McA po~> ~/7ce..J~ [CI'r g ~ OJLQ. cJerf~, i n C [QJ 81 0 /2>0f!:Jf!0nr.~ w /t le''LaiJlona.X. C~lVe/)-t·~ OVUL ckn~ ,'n C [et/ eJCi-en~(x.l~__-rac/to A c e~'-fL of CA- ~c~n_Q~~ ~ Q

i' SS 3.ejd~~. pefLC~C2- e:-,~ec({ i/ .

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" - \

IAnwveJL. - Y€ ~ .

D;,f~·cu/'ef:ie?::::.: t) If· ilL /ss no r110~ CA.- i/Ou(l7 G~p

tk ~r(J,-~ c9f J;''8:.con£lnf/l~fW7dior}~e. ~ ~ CL ·fUJ?d/~ VVYII'C:h;' '>!::,. no-t. '2:;<?-ICO on ~ anliniiefj tyl~ p",,!A3 )2;OJV€.. CZ'CJZD Lp- n,~

~e~i~ of Lp ([0; G 1) CULCL. oIe.j2l1r;ecA

e>n11 up' Q "2= rne~". ~. '"2.) TIk R1f'.n7eAnn.. In-f.J2~ l~ ncrtWf-fiuentf=- tk ~ ~n,.iicn. ;f 'Wch fund,<3i'L-~ vve... rJ-eed eL CT1--eC;G~ 'lh~ QVZo(Le~,~ ~ / f1 i,es.rz.et--6 iC)n .

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