APEIROSTIKOS LOGISMOS ApI_katav.pdf

:

2005-2006

1 2 3 4 Bolzano-Weierstrass 5 6 7 8 9 10

1 Algebrikc idithtec tou R

R : To snolo twn pragmatikn arijmnUposnola tou R

N = {0, 1, 2, . . .} : oi fusiko arijmo, N = {1, 2, . . .} = N \ {0}.

Z = N (N) oi akraioi arijmo.

Q = {mn: m,n Z, n 6= 0} = {m

n: m Z, n N} : oi rhto arijmo.

Oi idithtec twn prxewn kai thc ditaxhc:1

(1) 0 : a+ 0 = a a R (5) 1 6= 0 : a.1 = a a R(2) a (a) : a+ (a) = 0 (6) a 6= 0 a1 : a.a1 = 1

(3) a, b, c : a+ (b+ c) = (a+ b) + c (7) a, b, c : a.(b.c) = (a.b).c(4) a, b : a+ b = b+ a (8) a, b : a.b = b.a

(9) a, b, c : a.(b+ c) = a.b+ a.c

(10) a, b, c : (a > b) kai (b > c) a > c(11) a, b : a = b a > b a b a+ c > b+ c(13) a, b, c : (a > b) kai (c > 0) a.c > b.c

H sqsh a b shmanei: a b.

'Askhsh 1.1 'Estw a Z.O a enai rtioc o a2 enai rtioc.

'Askhsh 1.2 Den uprqei rhtc2 arijmc q ste q2 = 2.

1Upenjumzoume touc sumbolismoc: : gia kje, : uprqei2Ja dome argtera ti uprqei pragmatikc arijmc x ste x2 = 2. Proc to parn

parathrome ti h parxh ttoiou x den mpore na apodeiqje qrhsimopointac mnon ticidithtec P1 - P13.

1

Parathrseic 1.3 (i) Prsjesh anisottwn kat mlh:

a, b, c, d R : an a > b kai c > d tte a+ c > b+ d

(gia thn apdeixh, qrhsimopohse thn (P12) kai thn (P10)). Bebawc denmporome genik na afairome anisthtec kat mlh: p.q. 3 > 2 kai 5 > 1all 3 5 2 1.(ii) Den mporome en gnei na pollaplasizoume anisthtec kat mlh: p.q.3 > 2 kai 1 > 2 all 3(1) (2)(2). 'Omwc

a, b, c, d R : an a > b > 0 kai c > d > 0 tte ac > bd.

Prgmati, ap thn a > b afo c > 0 lgw thc (P13) qoume ac > bc kai apthn c > d afo b > 0 qoume bc > bd gia ton dio lgo. Sunepc ac > bc kaibc > bd ra ac > bd ap thn (P10).

(iii) 'Estw a, b R \ {0} omshmoi.

An a > b tte1

a 0 qoume kai 1ab

> 0 (giat?) kai sunepc ap thn a > blgw thc (P13) qoume a

ab> b

abdhlad 1

b> 1

a.

To sumprasma den isqei pnta an oi a kai b enai etershmoi. P.q. 3 > 2all 1

3 12 .

Askseic 1.4 (i) 'Estw a, b, x, y R me b > 0, y > 0. An ab< x

ytte (a)

ay < bx kai (b) ab< a+x

b+y< x

y.

(ii) 'Estw a R, a > 0 me a2 > 2. Jtoume b = a2+ 1

a. Tte b < a kai b2 > 2.

Lsh thc (ii) Ap th sqsh a2 > 2, afo a > 0 prokptei (pollaplasi-zontac me 1

2a) ti a

2> 1

akai sunepc

b =a

2+

1

a 0

2

(diti an a2 1

a= 0 ja eqame a2 = 2).

(iii) To snoloX = {x > 0 : x2 > 2}

den qei elqisto stoiqeo.

Lsh thc (iii) An doje na opoiodpote a X mporome na brome (gnh-swc) mikrtero stoiqeo tou X, p.q. to b = a

2+ 1

a, pwc petai ap thn (ii).

Sunepc to X den qei elqisto stoiqeo.

(iv) To snoloY = {x Q+ : x2 2}

den qei elqisto stoiqeo.

(v) An a, b R tte a2+b22

(

a+b2

)2. Pte isqei isthta?

(vi) An a, b R kai ab > 0 tte ab+ b

a 2.

(vii) An c > 0 kai n = 1, 2, . . . tte c2n + c2n1 + . . .+ c2 + c+1 (2n+1)cn.

Lsh thc (vii) Jtontac b = 1 sthn (vi) qoume a+ 1a 2 gia kje a > 0,

opte jtontac diadoqik a = c, met a = c2, . . . , a = cn qoume

c+1

c 2, c2 + 1

c2 2, . . . , cn + 1

cn 2 .

Prosjtontac kat mlh prokptei

cn + cn1 + . . .+ c+1

c+

1

c2+ . . .+

1

cn 2n

ra

cn + cn1 + . . .+ c+ 1 +1

c+

1

c2+ . . .+

1

cn 2n+ 1

kai pollaplasizontac ta do mlh me ton jetik arijm cn prokptei h zh-tomenh.

'Askhsh 1.5 (i) Den uprqei elqistoc jetikc arijmc: An 0 a kai, giakje > 0, isqei a < , tte a = 0.

(ii) An a b+ gia kje > 0 tte a b.(iii) An a < b+ gia kje > 0 enai aljeia ti a < b? ti a b?

3

Lsh (i) An a > 0 den mpore na isqei h sqsh a gia la ta jetik > 0: paradegmatoc qrin gia = a

2den isqei.

(ii) An den isqei to sumprasma, an dhlad a > b, tte h sqsh a b + den mpore na alhjeei gia kje > 0: p.q. gia = ab

2den isqei3.

(iii) Sto prto erthma h apnthsh enai arnhtik: mpore na qoume a = b.Sto detero erthma h apnthsh enai jetik ap thn prohgomenh skhsh.

Orismc 1.1 An a R, jtoume

|a| =

a an a 0

a an a < 0

Paratrhsh 1.6 Gia kje a R kai b 0

|a| b b a b dhl. b a kai a b|a| b a b b a

Paratrhsh 1.7 Gia kje a, b R

|a+ b| |a|+ |b| (trigwnik anisthta)| |a| |b| | |a b|| |a| |b| | |a+ b|

Apdeixh (i) Oi |a| a |a| kai |b| b |b| dnoun |a||b| a+b |a| + |b| ra (|a| + |b|) a + b |a| + |b| ra |a + b| |a| + |b| (ap thnprohgomenh Paratrhsh).

(ii) 'Estw c = |a| |b|. Epeid a = (a b) + b ap thn trigwnik anisthtaqw

|a| = |(a b) + b| |a b|+ |b|,ra c = |a| |b| |a b|. Omowc b = (ba)+a ra |b| |ba|+ |a|, opte

c = (|a| |b|) = |b| |a| |b a| = |a b|.

Dexame ti c |a b| kai c |a b|, ra |c| |a b|.(iii) Sthn (ii) ble b sth jsh tou b.

3giat a (b + ) = ab2 > 0 dhlad a > b + .

4

'Askhsh 1.8

|a+ b|1 + |a+ b|

|a|+ |b|1 + |a|+ |b|

|a|1 + |a|

+|b|

1 + |b|

Lsh H prth anisthta prokptei wc exc

|a+ b| |a|+ |b| (prosjtw |a+ b|(|a|+ |b|))

|a+ b|+ |a+ b|(|a|+ |b|) (|a|+ |b|) + (|a|+ |b|)|a+ b|

|a+ b|(1 + |a|+ |b|) (|a|+ |b|)(1 + |a+ b|)

|a+ b|1 + |a+ b|

|a|+ |b|1 + |a|+ |b|

.

Gia th deterh anisthta, jtontac x = |a| kai y = |b|, qw

x+ y

1 + x+ y=

x

1 + x+ y+

y

1 + x+ y

x1 + x

+y

1 + y

afo x 0 kai y 0.

Upenjmish:

Jerhma 1.9 (Majhmatik epagwg) 'Estw ti se kje fusik a-rijm n antistoiqe ma prtash P (n) pou afor ton n.Upojtoume ti ikanopoiontai oi exc do propojseic:

(i) H P (0) alhjeei,

KAI

(ii) gia kje fusik arijm m, an h P (m) alhjeeitte h P (m+ 1) alhjeei.

Tte h P (n) alhjeei gia kje fusik arijm n.

5

Pardeigma Na deiqje ti

1 + 2 + 3 + . . .+ n =n(n+ 1)

2

Apdeixh 'Estw P (n) h prtash

1 + 2 + 3 + . . .+ n =n(n+ 1)

2.

Prpei na dexoume ti h P (n) alhjeei gia kje n N.H prtash P (0) enai h 0 = 0: alhjc.

Isqurismc: P (k) P (k + 1). Prgmati:an 1 + 2 + 3 + . . .+ k = k(k+1)

2tte

1 + 2 + 3 + . . .+ k + (k + 1) =k(k + 1)

2+ (k + 1) =

(k + 2)(k + 1)

2

pou enai h P (k + 1).

Ma llh morf thc majhmatikc epagwgc enai h akloujh:

Jerhma 1.10 (Deterh morf thc epagwgc) 'Estw ti se kjefusik arijm n antistoiqe ma prtash P (n), pou afor ton n.Upojtoume ti ikanopoiontai oi exc do propojseic:

(i) H P (0) alhjeei,

KAI

(ii) gia kje fusik arijm m, an alhjeoun oi P (0), P (1), . . . , P (m), tteh P (m+ 1) alhjeei.

Tte h P (n) alhjeei gia kje fusik arijm n.

Pardeigma 'Estw (an)nN h akolouja Fibonacci pou orzetai ap ticsqseic:

a0 = 1, a1 = 1

an+2 = an+1 + an, n = 0, 1, 2, ...

Tte

an =n+1 ()(n+1)

5, n N

6

pou =1 +

5

2h qrus tom4.

Parathrome prta ti 2 = + 1 kai sunepc

1 = 1 = 15

2.

Gia n = 0 qoume

()15

=15

(1 +

5

2 1

5

2

)= 1 : alhjc.

Gia n = 1 qoume5

2 ()25

=15

(6 + 2

5

4 6 2

5

4

)= 1 : alhjc.

'Estw tra m N, m 2. Upojtoume ti h sqsh

an =n+1 ()(n+1)

5

alhjeei gia kje fusik n = 0, 1, . . . ,m 1 kai ja dexoume ti alhjeei gian = m. 'Eqoume

am = am1 + am2 =m ()m

5+

m1 ()m+15

=m1( + 1) ()m(1 )

5

=m1( 2) ()m(1)

5=

m+1 ()(m+1)5

.

Suqn sunantme protseic pou alhjeoun gia kje fusik arijm megal-tero ap kpoion no N, qi mwc endeqomnwc gia touc prohgomenouc.

4Gia tic angkec tou paradegmatoc deqmaste (kai ja apodexoume sto epmeno kef-laio) ti uprqei monadikc jetikc pragmatikc arijmc

5 me thn idithta (

5)2 = 5.

5O a1 den mpore na prokyei ap ton anadromik tpo, giat autc orzei ton amqrhsimopointac kai touc do prohgomenouc rouc.

7

Prisma 1.11 'Estw ti se kje fusik arijm n antistoiqe ma prtashP (n) kai stw no N. Upojtoume ti ikanopoiontai oi exc do propoj-seic:

(i) H P (no) isqei.

kai

(ii) Gia kje n N, n no, an h P (n) isqei, tte isqei h P (n+ 1).Tte h P (n) isqei gia kje fusik arijm n no.

Pardeigma 1.12 H anisthta 2n n2 isqei gia kje n 4, qi mwcgia n = 3.

Apdeixh Ed P (n) enai h prtash 2n n2. H P (3) den alhjeei, afo23 < 32.

H P (4) : 24 42 enai alhjc.Mnei na apodexoume thn sunepagwg

2n n2 2n+1 (n+ 1)2

gia kje n 4. Prgmati, an 2n n2 tte

2n+1 (n+ 1)2 = 2.2n (n2 + 2n+ 1) 2.n2 (n2 + 2n+ 1)= n2 2n 1 = (n 1)2 2 0

efson n 4.

To diwnumik anptugma

(a+ b)0 = 1a+ b = a+ b

(a+ b)2 = a2 + 2ab+ b2

(a+ b)3 = a3 + 3a2b+ 3ab2 + b3

(a+ b)4 = a4 + 4a3b+ 6a2b2 + 4ab3 + b4

(a+ b)5 = a5 + 5a4b+ 10a3b2 + 10a2b3 + 5ab4 + b5

(a+ b)6 = a6 + 6a5b+ 15a4b2 + 20a3b3 + 15a2b4 + 6ab5 + b6

8

Oi suntelestc sqhmatzoun to legmeno trgwno tou Pascal:

11 1

1 2 11 3 3 1

1 4 6 4 11 5 10 10 5 1

1 6 15 20 15 6 1

Kje suntelestc enai to jroisma twn do suntelestn pou brskontaiakribc ap pnw tou. Aut h paratrhsh enai h ida gia thn epagwgikapdeixh pou akolouje.

Ja qreiasjon oi

Sumbolismo

0! = 1, n! = 1.2. . . . .n (dhl. n! = n(n 1)!)(n

k

)=

n!

k!(n k)!=

n(n 1) . . . (n k + 1)k!

nk=0

ak = a0 + a1 + . . .+ an.

Parathrome ti to jroisma a0 + a1 + . . .+ an mpore epshc na grafte

nm=0

am =n+1j=1

aj1.

H isthta twn do autn parastsewn mpore na prokyei kai me thn {allagmetablhtc} j = m+ 1.

Prtash 1.13 (Diwnumik anptugma) An a, b R kai n N, n 1, tte

(a+ b)n =n

k=0

(n

k

)ankbk (: P (n)).

9

Shmewsh: Gia n 2 qoume

(a+ b)n =an + nan1b+n(n 1)

2!an2b2 +

n(n 1)(n 2)3!

an3b3 + . . .

+n(n 1)

2!a2bn2 + nabn1 + bn.

H apdeixh gnetai me epagwg:

Gia n = 1 h P (1) enai h a+ b = a+ b pou profanc alhjeei.

Ja dexoume thn sunepagwg P (n) P (n+ 1):An

(a+ b)n =n

k=0

(n

k

)ankbk.

tte

(a+ b)n+1 = (a+ b)n

k=0

(n

k

)ankbk = a

nk=0

(n

k

)ankbk + b

nk=0

(n

k

)ankbk

=n

k=0

(n

k

)an+1kbk +

nm=0

(n

m

)anmbm+1

= an+1(k=0)

+n

k=1

(n

k

)an+1kbk +

n1m=0

(n

m

)anmbm+1 + bn+1

(m=n)

= an+1 +n

k=1

(n

k

)an+1kbk +

nk=1

(n

k 1

)an(k1)bk

(k=m+1)

+ bn+1

= an+1 +n

k=1

[(n

k

)+

(n

k 1

)]an+1kbk + bn+1.

10

All(n

k

)+

(n

k 1

)=

n!

k!(n k)!+

n!

(k 1)!(n k + 1)!

=n!

k(k 1)!(n k)!+

n!

(k 1)!(n k + 1)(n k)!

=n!(n k + 1)

k(k 1)!(n k)!(n k + 1)+

n!k

k(k 1)!(n k + 1)(n k)!

=n!(n k + 1 + k)k!(n k + 1)!

=(n+ 1)!

k!(n+ 1 k)!=

(n+ 1

k

)ra

(a+ b)n+1 = an+1+n

k=1

(n+ 1

k

)an+1kbk + bn+1 =

n+1k=0

(n+ 1

k

)an+1kbk. 2

Prtash 1.14 (Anisthta Bernoulli) An x > 1,

(1 + x)n 1 + nx gia kje n N.

Apdeixh H anisthta gia n = 0 enai profanc: 1 = 1.

Deqnoume to epagwgik bma:

An (1+x)n 1+nx tte (1+x)(1+x)n (1+x)(1+nx) (efson 1+x > 0)ra

(1 + x)n+1 = (1 + x)(1 + x)n (1 + x)(1 + nx)= 1 + (n+ 1)x+ nx2 1 + (n+ 1)x. 2

Paratrhsh Gia x 0 h anisthta Bernoulli enai mesh sunpeia toudiwnumiko anaptgmatoc, afo loi oi prosjetoi tou enai mh arnhtiko.

Prtash 1.15 An n N, n 1 kai oi b1, . . . bn enai jetiko arijmo meginmeno b1b2 . . . bn = 1, tte b1 + b2 + . . .+ bn n.

11

Apdeixh Me epagwg sto pljoc n twn arijmn pou emfanzontai. Gian = 1 h anisthta enai tetrimnh: 1 1.Upojtoume loipn ti gia kje m-ada jetikn arijmn d1, . . . dm me ginmenod1d2 . . . dm = 1 isqei h anisthta

d1 + d2 + . . .+ dm m

kai jloume na dexoume ti an dojon m + 1 jetiko arijmo b1, . . . bm+1 meginmeno b1b2 . . . bm+1 = 1 tte ja isqei h

b1 + b2 + . . .+ bm+1 m+ 1 .

Parathrome ti, an b1 = b2 = . . . = bm+1, h anisthta isqei profanc. Anqi, diatssontac touc arijmoc b1, . . . , bm+1 kat axousa seir megjouc(prgma pou den allzei to jroism touc), parathrome ti anagkastikb1 < 1 < bm+1 (allic to ginmeno lwn touc den ja tan 1).

An jewrsw thn m-ada jetikn arijmn

b1.bm+1 , b2, . . . , bm

pou qoun ginmeno 1, ap thn epagwgik upjesh ja qw

b1.bm+1 + b2 + . . .+ bm m. (1)

'Omwc ap tic sqseic b1 < 1 < bm+1 petai ti (bm+1 1)(1 b1) > 0 dhladb1 + bm+1 > bm+1.b1 + 1 kai sunepc

b1 + bm+1 + b2 + . . .+ bm > b1.bm+1 + 1 + b2 + . . .+ bm(1)

m+ 1 .

O gewmetrikc msoc miac n-dac a1, . . . , an jetikn arijmn enai o mo-nadikc jetikc arijmc pou ikanopoie n = a1a2 . . . an. O arijmc autcsumbolzetai = n

a1.a2 . . . an. Thn parxh kai monadiktht tou ja apode-

xoume sto epmeno Keflaio.

Prtash 1.16 (Anisthta arijmhtiko-gewmetriko msou)An n N, n 1 kai oi a1, . . . , an enai jetiko arijmo, tte

a1 + a2 + . . .+ ann

na1.a2 . . . an . (2)

12

Apdeixh An jsw

bk =ak

na1.a2 . . . an

(k = 1, 2, . . . , n), parathr ti oi bk enai jetiko arijmo me ginmeno

a1na1.a2 . . . an

. . .an

na1.a2 . . . an

=a1a2 . . . an

( na1.a2 . . . an)n

= 1.

'Ara ap thn prohgomenh Prtash ja qoume

b1 + b2 + . . .+ bn n , (3)

isodnamaa1 + a2 + . . .+ an

na1.a2 . . . an

n

pou enai isodnamh me thn apodeikta. 2

Paratrhsh 1.17 'Opwc dh parathrsame, an oi arijmo a1, . . . an enailoi soi, tte h anisthta (2) enai sthn pragmatikthta isthta. An den enailoi soi, tte pwc prokptei ap thn apdeixh, h anisthta enai gnsia (ditib1 + bm+1 > bm+1.b1 + 1). Sunepc:

Sthn anisthta arijmhtiko-gewmetriko msou isqei isthta an kai mnonan a1 = a2 = . . . = an.

13

2 H Plhrthta tou R

Orismc 2.1 'Estw A R kai a R.

O a lgetai na nw frgma tou A an kje x A ikanopoie x a.To A lgetai nw fragmno an qei kpoio nw frgma.

O a lgetai na ktw frgma tou A an kje x A ikanopoie x a.To A lgetai ktw fragmno an qei kpoio ktw frgma.

To A lgetai fragmno an enai kai nw fragmno kai ktw fragmno.

Parathrseic 2.1 (i) 'Enac arijmc m lgetai mgisto1 stoiqeo tou A(grfoume m = maxA) an to m ankei sto A kai enai enai nw frgma touA. An to A qei mgisto stoiqeo, tte enai nw fragmno. All na snolompore na enai nw fragmno qwrc na qei mgisto stoiqeo.

(ii) To A enai fragmno an kai mnon an uprqei c 0 ste |x| c gia kjex A.(iii) An o a enai na nw frgma tou A, tte kje b a enai epshc na nwfrgma tou A.

Orismc 2.2 'Estw A R kai a R. O a lgetai elqisto nwfrgma (supremum) tou A an enai to mikrtero ap ta nw frgmata touA. Dhlad

(i) O a enai na nw frgma tou A kai

(ii) Kje nw frgma b tou A ikanopoie b a.

Grfoume a = supA.

Orismc 2.3 'Estw A R kai b R. O b lgetai mgisto ktwfrgma (infimum) tou A an enai to megaltero ap ta ktw frgmatatou A.

Grfoume b = inf A.

1'Enac arijmc lgetai elqisto stoiqeo tou A (grfoume = minA) an Akai enai ktw frgma tou A.

1

Parathrseic 2.2 (a) To supA, an uprqei, enai monadik. To dio kaito inf A.

(b) 'Ena mh ken kai nw fragmno snolo A R mpore na qei na mhnqei mgisto stoiqeo. An to A qei mgisto stoiqeo, stw m, tte bbai-a (to supA uprqei kai) m = supA. En gnei mwc to supA den ankeikatangkhn sto A. Gia pardeigma to snolo A = {x R : 0 x < 2} qeisupremum, to 2.

To den qei supremum, giat kje arijmc enai nw frgma tou. Prgmati,an to a den tan nw frgma tou , ja prepe na uprqei x ste x > a.All to den qei stoiqea!To snolo A = {x R : x2 < 2} enai nw fragmno, paradegmatoc qrinap to 3. Prgmati, kje x A ikanopoie x 3 (giat, an uprqe x A mex > 3 tte ja eqame x2 > 32 = 9 > 2). Gia ton dio lgo, kje pragmatikcarijmc b 0 me b2 > 2 enai nw frgma tou A, paradegmatoc qrin to 1, 5(giat (1, 5)2 = 2, 25 > 2, to 1, 42 (giat (1, 42)2 = 2, 0164 > 2), kai otwkajexc.Uprqei mwc elqisto nw frgma?

Diaisjhtik, fanetai ti h apnthsh enai katafatik. Aut mwc denmpore na apodeiqje mnon ap tic algebrikc idithtec P1 - P13 twnpragmatikn arijmn. Prpei na prosteje mia akma, kai mlista krsimh,idithta tou sunlou twn pragmatikn arijmn:

P 14: Idithta plhrthtac tou sunlou twn pragmatikn a-rijmn Kjemh ken kai nw fragmno uposnolo tou R qei elqistonw frgma (supremum).

Prtash 2.3 Uprqei monadikc jetikc pragmatikc arijmc a stea2 = 2.

Apdeixh 'Estw

A = {x R : x > 0 kai x2 < 2}.

To A den enai ken (p.q. 1 A) kai enai nw fragmno (p.q. ap to 2).Ap thn 14, qei elqisto nw frgma, stw a. Afo 1 A, qw a 1.Ja dexw ti a2 = 2, apokleontac tic periptseic a2 > 2 kai a2 < 2 :

2

1) Ac upojsoume ti a2 > 2. Ja epilxw v (0, a) ste (a v)2 > 2.'Eqoume

(a v)2 = a2 2av + v2 > a2 2av.

Sunepc arke na epilxw to v > 0 ste

a2 2av 2 a2 2 2av a>0 a2 22a

v .

Epilgw loipn v = a222a

. Efson a v < a, o arijmc a v den enai nwfrgma tou A. Epomnwc uprqei x A me x > a v. All a v > a

2> 0,

opte x2 > (a v)2. Afo (a v)2 > 2, qoume x2 > 2, topo.

2) Ac upojsoume ti a2 < 2. Ja br kpoio u > 0 ste (a + u)2 < 2.Mpore mlista na breje ttoio me 0 0 u 2 a2

2a+ 1.

An loipn epilxw nan jetik arijm u me u 0

(i) gia kje x A isqei x < a+ kai(ii) uprqei toulqiston na y A ste a < y .

Apdeixh 'Estw ti a = supA. Jewr na > 0 (tuqao). To a enai nwfrgma tou A kai sunepc gia kje x A isqei x a, ra x < a + .Epshc to a den enai nw frgma tou A ra mporome na brome y Aste a < y (an den uprqe ttoio y, tte kje x A ja ikanopoiosex a , opte to a ja tan nw frgma tou A).'Estw antstrofa ti to a ikanopoie tic (i) kai (ii) gia kje jetik arijm. Tte ap thn (i) sumperanoume ti to a enai nw frgma tou A, giat anden tan ja uprqe x A ste a < x kai tte h (i) den ja tan alhjc p.q.gia = xa

2. H (ii) lei ti kannac arijmc a gnsia mikrteroc tou a

den mpore na enai nw frgma tou A, epomnwc to a enai to elqisto nwfrgma tou A. 2

2giat an tan perittc, stw m = 2k + 1, tte o m2 = 4k2 + 4k + 1 ja tan perittc

4

Paratrhsh 2.8 H prohgomenh paratrhsh qrhsimopoietai suqn me thnakloujh aplosterh morf:

'Estw A mh ken uposnolo tou R kai a R na nw frgma tou. Tte:To a enai to elqisto nw frgma tou A an kai mnon an gia kje > 0uprqei x A ste a < x.Mia efarmog thc Paratrhshc autc enai kai h akloujh:

Paratrhsh 2.9 Kje mh ken kai nw fragmno snolo akerawn arij-mn qei mgisto stoiqeo.

Apdeixh 'Estw A Z mh ken kai nw fragmno. Ap thn idithta thcplhrthtac tou R, uprqei to s = supA R. Prpei na dexoume ti s A.Efarmzontac thn Paratrhsh 2.8 gia = 1, brskoume x A ste s 1 0 uprqei n N ste 1n< .

Apdeixh Allic ja uprqe > 0 ste 1n gia kje n N, opte to N

ja tan nw fragmno (ap to 1).

5

Prisma 2.13 An a, b enai jetiko pragmatiko arijmo, uprqei n Nste na > b.

Apdeixh O arijmc baden enai nw frgma tou N.

Prtash 2.14 (Puknthta twn rhtn sto R) Anmesa se duo (dia-foretikoc) pragmatikoc arijmoc uprqei pnta rhtc: An a, b R kai a < buprqei c Q ste a < c < b.Apdeixh Anmesa se duo rhtoc arijmoc uprqei pnta rhtc, p.q. o a+b

2.

Ti gnetai an oi a, b den enai kai oi do rhto?

Upojtw ti 0 < a < b. (Oi llec periptseic afnontai ston anagnsth).Jlw na br m,n N ste a < m

n< b.

'Estw n N, n 6= 0. To snolo {k N : kn< b} den enai ken (periqei

to 0), ra qei mgisto stoiqeo (Paratrhsh 2.9), stw m. To m enai tomegaltero k pou ikanopoie thn anisthta k

n< b, ra to m + 1 den thn

ikanopoie. 'Eqoume loipn

m

n< b m+ 1

n.

Pc mwc ja exasfalsw ti ja isqei kai h a < mn? Epilgontac prta

arket mikr n: Parathr ti, an a mn, tte ja qw 1

n= m+1

n m

n b a.

An loipn epilxw ex arqc to n ste 1n b (b a) = a. 2

Paratrhsh 2.15 Anmesa se duo (diaforetikoc) pragmatikoc arij-moc uprqei pnta rrhtoc.

Apdeixh Prgmati, an a < b tte2a 0, tte (+) = (+) = +kai () = () =

An < 0, tte (+) = (+) = kai () = () = +

(+) + (+) = + , () + () = (+) (+) = + , () () = +,(+) () = () (+) = .

Den orzontai oi parastseic(+) + () , () + (+) ,0 (+) , (+) 0 , 0 () , () 0++

,+

,+

,

.

Me touc sumbolismoc autoc, an na mh ken snolo A R den enai nwfragmno (opte to supA den uprqei sto R), grfoume suqn supA = +,ki an den enai ktw fragmno, grfoume inf A = .

7

3 Akoloujec

Orismc 3.1 Akolouja (pragmatikn arijmn) enai mia anti-stoqish 1 a1, 2 a2, . . . , n an, . . . twn fusikn arijmn 1, 2, . . . , n, . . .proc touc pragmatikoc arijmoc. Grfoume (an) (an)nN (a1, a2, . . .), endeqomnwc (a0, a1, . . .).

Dhlad akolouja pragmatikn arijmn enai mia sunrthsh me pedo orismoto snolo N twn fusikn arijmn ( to N\{0}) kai timc sto R. Shmeinoumeth diafor anmesa sthn akolouja (a0, a1, . . .) kai sto snolo {a0, a1, . . .} ={an : n N} twn timn thc. Paradegmatoc qrin h akolouja (1)n =(1,1, 1,1, . . .) qei snolo timn to {1, 1}. Epshc oi akoloujec (an)kai (bn) pou

(an) = (1

2,1

4,1

6,1

8,1

10,1

12, . . . , )

(bn) = (1

2,1

2,1

2,1

4,1

2,1

6. . . , )

dhlad an =12n

gia kje n N, n 1 kai bn ={

1n

n rtioc12

n perittcenai

pol diaforetikc, qoun mwc to dio snolo timn { 12m

: m = 1, 2, . . .}.

Orismc 3.2 'Estw (an) akolouja kai a R. Lme ti h (an) sugklneisto a kai grfoume an a lim

nan = a kajc n lim

nan = a an:

gia kje > 0 uprqei no N (pou exarttai ap to )ste gia kje n no na isqei |an a| < .

Sumbolik: > 0 no N : n no |an a| < .

Lme ti h (an) sugklnei an uprqei a R ste limnan = a. Allic lme

ti apoklnei.

Dhlad h (an) sugklnei sto a R an OPOIO (osodpote mikr) > 0 kai nadoje, KAPOIO telik tmma (ano , ano+1, ano+2, . . .) brsketai OLOKLHROsthn perioq (a , a+ ).

1

Isodnama, h (an) sugklnei sto a R an gia kje > 0 h sqsh |ana| den mpore na isqei gia peiro pljoc deiktn n, dhlad

gia KAJE > 0 to snolo twn deiktn {n N : |an a| }enai peperasmno.

Epomnwc, h (an) DEN sugklnei sto a R angia KAPOIO > 0 to snolo twn deiktn {n N : |an a| }enai APEIRO

isodnama UPARQEI > 0 ste KANENA telik tmma (an, an+1, an+2, . . .)na MHN brsketai OLOKLHRO sthn perioq (a , a+ ), dhlad kpoiocroc thc akoloujac (an, an+1, an+2, . . .) na brsketai ektc thc perioqc(a , a+ ). Dhlad

Paratrhsh 3.1 H akolouja den sugklnei sto a an uprqei > 0 stegia kje n N na uprqei m n ste |am a| .

Paradegmata 3.2 H akolouja (an) pou an =1ntenei sto 0. To dio kai

h akolouja (bn) pou bn =

nn n 1023

80n

n > 1023.

H akolouja (cn) pou cn = (1)n den sugklnei, all h (dn) pou dn = (1)n

n

tenei sto 0.

H akolouja (fn) pou fn = n2 den sugklnei se pragmatik arijm. Ote h

akolouja (gn) pou gn =

n2 n rtioc

1n

n perittc

Apodexeic Efson to snolo N twn fusikn arijmn den enai nw frag-mno1, gia kje > 0 uprqei no N ste 1no < . Den arke mwc aut giana exasfalsoume ti 1

n 0: prpei loi oi roi ap ton no-ost kai pra na

ikanopoion thn anisthta. Aut mwc enai aljeia (sthn perptws mac):an n no tte 1n

1no

< .

Gia thn (bn), arke, gia kje > 0, na prw no > max{80 , 1023}. Tte, an

n no ja qw |bn 0| = 80n 80no

< .

1Paratrhse ti ed qrhsimopoietai h Arqimdeia idithta tou N, pou me th seir thcsthrzetai sthn plhrthta tou R.

2

An h (cn) eqe rio kpoion arijm c R, tte ja uprqe no N ste|cn c| < 1/2 tan n no (efrmosa ton orism me = 1/2). Tte mwc japrepe na isqei |cn+1 cn| |cn+1 c| + |c cn| < 1 tan n no, prgmapou den isqei afo |cn+1 cn| 1 gia kje n.To dio epiqerhma deqnei ti kai oi llec do akoloujec den sugklnoun,giat |fn+1 fn| = 2n+ 1 1 kai |gn+1 gn| 1 gia kje n > 1. 2

Parathrseic 3.3 (i) An an a tte |an| |a|. To antstrofo mwcden isqei en gnei, tan a 6= 0 ('Askhsh!).(ii) Enai meso ap ton orism thc sgklishc ti h sqsh an a isodunameme thn |an a| 0.(iii) H sgklish kai h tim tou orou miac akoloujac (an) exartntai mnonap na {telik tmma} (an)nno thc akoloujac: p.q. pwc edame sta proh-gomena Paradegmata, h akolouja (bn) qei to dio rio me thn

(80n

), giat

oi do akoloujec tautzontai gia kje n > 1023.

Epshc an to rio limn an uprqei, tte uprqei kai to rio limn an+1 kai enaisa. Geniktera, an k Z kai bn = an+k (n k), h akolouja (an) sugklneisnan arijm a an kai mnon an h akolouja (bn)nk sugklnei sto a. (Hapdeixh enai mesh ap ton orism thc sgklishc kai afnetai c skhsh.)

Anmesa stic akoloujec pou apoklnoun, mia eidik klsh axzei na apomo-nwje: ekenec pou, katapluto tim, gnontai {aujareta meglec} (pwc h(fn) sto teleutao pardeigma):

Orismc 3.3 Lme ti h akolouja (an) tenei sto + an gia kje M R uprqei no N (pou exarttai ap to M) ste an > M gia kje n no.Lme ti h (an) tenei sto an gia kje M R uprqei no N (pouexarttai ap to M) ste an < M gia kje n no.(Tonzoume ti oi akoloujec autc apoklnoun)!

Prtash 3.4 (Monadikthta orou) An an a kai an b, ttea = b.

Apdeixh 'Estw a 6= b. Afo an a, gia kje > 0, ra kai gia = |ab|2 ,uprqei no N ste

3

|an a| < tan n no. (1)

Afo an b, gia to dio uprqei n1 N ste

|an b| < tan n n1. (2)

Epomnwc an n no kai n n1 (p.q. n = max{no, n1}) ikanopoiontaitautqrona kai oi do anisthtec. Tte mwc

|a b| |a an|+ |an b| < 2 = |a b|

prgma adnato. 2

Orismc 3.4 Lme ti h akolouja (an) enai nw fragmnh (ktwfragmnh, fragmnh) an to snolo {an : n N} twn rwn thc enainw fragmno (ktw fragmno, fragmno).

Dhlad h (an) enai:nw fragmnh an uprqei M > 0 ste an M gia kje n N,ktw fragmnh an uprqei M R ste an M gia kje n N, kaifragmnh an enai nw kai ktw fragmnh, dhl. isodnama an uprqei K > 0ste |an| K gia kje n N.

Prtash 3.5 Kje sugklnousa akolouja enai fragmnh (to antstrofomwc den isqei en gnei).

Apdeixh 'Estw ti an a. Tte (gia kje > 0, ra kai gia = 1)uprqei no N ste |an a| < 1, ra |an| < |a| + 1, tan n no. Dhladto snolo

{ano , ano+1, ano+2, . . .} = {an : n no}enai fragmno (p.q. ap ton arijm |a|+ 1). 'Omwc to {uploipo} snolo

{a1, a2, . . . , ano1}

enai peperasmno, sunepc kai aut fragmno (kpoioc ap touc rouc au-toc, stw o |ak|, pou k < no, ja qei thn mgisth apluth tim). 'Ara kaih nwsh twn do sunlwn, dhlad to {an : n N}, ja enai fragmno (p.q.ap to max{|a|+ 1, |ak|}).Pardeigma fragmnhc akoloujac pou den sugklnei enai h ((1)n) (dec pa-rdeigma 3.2). 2

4

Prtash 3.6 An an a kai bn b, tte:(1) an + bn a+ b(2) an a gia kje R.(3) an.bn a.b(4) An b 6= 0, tte uprqei no ste bn 6= 0 gia kje n no kai h akolouja(anbn

)nno sugklnei stoa

b.

Apdeixh tou (1) 'Estw > 0. 'Eqoume

|(an + bn) (a+ b)| |an a|+ |bn b|.

Gia na petqoume to |(an + bn) (a + b)| na enai mikrtero ap , ARKEIna knoume kje prosjeto mikrtero ap /2. 'Estw n1 ste |an a| 0. 'Eqoume

|anbn ab| = |anbn anb+ anb ab| |anbn anb|+ |anb ab|= |an|.|bn b|+ |an a|.|b| M.|bn b|+ |an a|.|b|

'Estw n1 ste |bn b| < /2M gia kje n n1 (uprqei ttoio n1 efsonbn b). 'Estw n2 ste |an a| < /2|b| gia kje n n2 (uprqei ttoion2 efson an a). Tte gia kje n megaltero so kai ap ta do (dhl.n max{n1, n2}) qoume M.|bn b| < /2 kai |an a|.|b| < /2, ra

|an.bn a.b| M.|bn b|+ |an a|.|b| < .

Apdeixh tou (2) 'Epetai ap to (3) jewrntac thn stajer akolouja(bn) me bn = gia kje n N.

Apdeixh tou (4) Paratrhse prta ti an x>0 kai |yx|< x2tte |y|> x

2.

Efson bn b, uprqei no ste gia kje n no na isqei |bn b| < |b|2 .Tte

||bn| |b|| |bn b| |b|2

5

(opte bn 6= 0 gia kje n no) kai 1bn 1b = b bnbbn

< |b bn||b| |b|2

= 2|b bn|

b2.

'Estw tra > 0. Epilgontac n1 ste n1 no kai 2 |bbn|b2 < gia kjen n1 (uprqei ttoio n1 efson bn b) qoume

1bn 1b < gia kjen n1.Dexame loipn ti h akolouja ( 1

bn)nno sugklnei sto

1b. Epomnwc, ap to

(3), h akolouja (anbn)nno sugklnei sto

ab. 2

Shmewsh To antstrofo thc Prtashc den isqei. Antiparadegmata: an =(1)n, bn = (1)n+1.

Paratrhsh 3.7 An an a, bn b kai an bn gia kje n tte a b.An mwc an < bn gia kje n den petai ti a 0. Brec n1 N ste

gia kje n n1 na isqei a < an < a+ . (3)

Brec n2 N stegia kje n n2 na isqei a < cn < a+ . (4)

Tte gia kje n max{no, n1, n2} isqoun oi do teleutaec anisthteckajc kai h an bn cn kai sunepc

a (3)< an bn cn

(4)< a+

ra |bn a| < gia kje n max{no, n1, n2}, prgma pou deqnei ti hakolouja (bn) sugklnei, kai mlista sto a. 2

Gia pardeigma, an |bn| cn gia kje n N kai cn 0, tte (|bn| 0 ra)bn 0. 'Etsi apodeiknetai p.q. ti 100n3 0, efson

1n 0 kai 100

n3 1

ntan

n 10.

6

'Askhsh 3.9 lim 1n= 0.

Apdeixh 'Estw > 0. Ja brw no ste1no

< , isodnama (afo no > 0)1no

< 2, dhlad no >12. Uprqei ttoio no efson to N den enai nw

fragmno2. Gia kje n no qw

1n 1

no< . 2

Prtash 3.10 (1) An a > 0, limna1/n = 1, (2) lim

nn1/n = 1.

Apdeixh (1) 'Estw prta a 1. Tte a1/n 1 gia kje n N, opte oarijmc bn = a1/n 1 enai mh arnhtikc. 'Eqw a = (1 + bn)n 1 + nbn apto diwnumik anptugma thn anisthta Bernoulli, ra, gia kje n 1,

0 bn a 1n

,

all a1n 0, ra bn 0 ap thn Prtash 3.8.

An a < 1 tte 1a> 1 kai ra 1

a1/n= ( 1

a)1/n 1.

(2) Gia n > 2 qw n1/n > 1, opte dn n1/n 1 > 0. Ap to diwnumikanptugma qw

n = (1 + dn)n = 1 + ndn +

n(n 1)2

d2n + . . .+ dnn

n(n 1)2

d2n

ra 1 n 12

d2n kai sunepc

0 < dn

2

n 1.

All

2n1 0 ap thn 'Askhsh 3.9, ra dn 0. 2

Ac jumhjome ti mia sunrthsh f : X R me pedo orismo X R lgetaiaxousa an gia kje x, y X me x < y isqei f(x) f(y). Afo miaakolouja enai mia sunrthsh me pedo orismo to N, h akolouja (an) ja

2Mlista to elqisto katllhlo no enai to [ 12 ] + 1, all den qei shmasa.

7

enai axousa an an am gia kje n,m N me n < m. Aut bebawcsunepgetai ti an an+1 gia kje n N. All kai antstrofa, an isqeih sqsh an an+1 gia kje n N, tte h (an) enai axousa. Prgmati, giakje n N kai k N qoume3 an an+k. Epomnwc an n,m N kai n < m,jtontac k = m n, qoume an an+k = am. Sunoyzoume:

Paratrhsh 3.11 Mia akolouja (an) pragmatikn arijmn enai:

axousa an an an+1 gia kje n N.

gnhswc axousa an an < an+1 gia kje n N.

fjnousa an an an+1 gia kje n N.

gnhswc fjnousa an an > an+1 gia kje n N.

montonh an enai axousa fjnousa.

gnhswc montonh an enai gnsia axousa gnsia fjnousa.

Jerhma 3.12Kje montonh kai fragmnh akolouja pragmatikn arijmn sugklnei.

An h (an) enai axousa kai nw fragmnh tte limn an = sup{an : n N},en an enai fjnousa kai ktw fragmnh tte limn an = inf{an : n N}.

Paratrhsh To antstrofo den isqei en gnei: mia sugklnousa ako-louja (enai bebawc fragmnh (Prtash 3.5) all) den enai anagkastik

montonh: pardeigma h

((1)n

n

).

Apdeixh Jewrmatoc Ac upojsoume ti h (an) enai fjnousa kai ktwfragmnh (h llh perptwsh apodeiknetai moia). To snolo A = {an :n N} enai mh ken kai ktw fragmno. Epomnwc ap thn idithta thcplhrthtac4 to A qei mgisto ktw frgma (infimum), stw a.

'Estw > 0. Afo o arijmc a + den enai ktw frgma tou A, uprqeikpoio ano A ste ano < a + . All h (an) enai fjnousa, epomnwc gia

3Aut apodeiknetai me epagwg sto k: Gia k = 1 h anisthta isqei ex upojsewc, kaian an an+j tte, efarmzontac pli thn upjesh gia ton fusik arijm n + j qoumean+j an+j+1 kai sunepc an an+j an+j+1.

4gia thn akrbeia, ap na meso prism thc

8

kje n no qoume an ano , ra an < a+ . Ap thn llh meri, to a enaiktw frgma tou A, ra an a > a gia kje n.Dexame ti gia kje > 0 uprqei no ste gia kje n no na isqeia < an < a+ . Sunepc an a. 2

'Askhsh 3.13 'Estw A R mh ken kai fragmno. Uprqei tte mia a-xousa akolouja (an) me an A gia kje n ste an supA kai mia fjnousaakolouja (bn) me bn A gia kje n ste bn inf A. Pte uprqei gnhswcaxousa akolouja stoiqewn tou A pou na sugklnei sto supA?

'Askhsh 3.14 Kje x R enai rio miac akoloujac rhtn (kai miac a-koloujac arrtwn). Mlista, uprqei mia gnhswc axousa kai mia gnhswcfjnousa akolouja rhtn pou na sugklnei sto x.

Pardeigma 3.15 'Estw a1 = 1 kai an+1 =1 + an. H (an) sugklnei

ston arijm =1 +

5

2(th qrus tom).

Apdeixh Ja dexoume prta, qrhsimopointac epagwg, ti h (an) enaiaxousa kai fragmnh. Gia na dexoume ti an+1 an gia kje n, parathromeprta ti a2 =

1 + 1 > 1 = a1. 'Estw tra m N. An am+1 am, qoume

am+2 =

1 + am+1 1 + am = am+1

kai to epagwgik bma apodeqjhke.

Ja dexoume tra ti 1 an 2 gia kje n. 'Eqoume katarqn 1 = a1 anafo h (an) enai axousa. Epshc, h anisthta am 2 isqei gia m = 1,afo a1 = 1. All

am 2 am+1 =1 + am

1 + 2 2

ra h anisthta isqei gia kje n.

Ap to Jerhma 3.12 petai tra ti h (an) sugklnei se kpoion arijma. Efson 1 an 2 gia kje n ja qoume 1 a 2. Pc mwcja prosdiorsoume thn tim tou a? Qrhsimopoiome thn anadromik sqshan+1 =

1 + an: epeid limn an+1 = a (Paratrhsh 3.3 (iii)), petai ti

a2 = limna2n+1 = lim

n(1 + an) = 1 + a

9

kai sunepc a2 a 1 = 0, prgma pou sunepgetai a = 1 +5

2, giat h

llh rza a =1

5

2thc exswshc den ikanopoie thn anisthta a 1.

Prtash 3.16 An 0 < a < 1, tte limn an = 0.

Prth apdeixh H akolouja (an) enai (gnhswc) fjnousa kai ktwfragmnh ap to 0. Ap to Jerhma 3.12, uprqei to b = limn an. Efsonan > 0 gia kje n, qoume b 0. Paratrhse ti oi akoloujec (an) kai(an+1) qoun to dio rio (Paratrhsh 3.3 (iii)). Sunepc

ba = (limnan)a = lim

nan+1 = lim

nan = b b(1 a) = 0.

All a 6= 1, ra b = 0.

Deterh apdeixh Efson 1a> 1, qoume 1

a 1 d > 0. Ap thn

anisthta Bernoulli to diwnumik anptugma qoume ti (1 + d)n 1 + ndopte

0 an = 1(1 + d)n

11 + nd

0

kai ra an 0 ap thn Prtash 3.8.

'Askhsh 3.17 'Estw an 0 gia kje n. An uprqei (0, 1) ste nan

gia kje5 n N, tte to rio limn an uprqei kai enai 0.

Apdeixh Efson 0 < < 1, to rio limn n uprqei kai enai 0. All

0 an n gia kje n N,

ra to rio limn an uprqei kai enai 0.

Paratrhsh 3.18 Enai krsimo na mpore na breje frgma gnsia mi-krtero ap 1. Paradegmatoc qrin an an =

1n1/n

, tte nan 1 gia kje

n N, all h (an) den sugklnei sto 0 (Prtash 3.10).5'Opwc fanetai ap thn apdeixh, arke h anisthta na ikanopoietai {telik}, na uprqei

dhlad kpoio no N ste nan gia kje n no.

10

Lmma 3.19 (i) 'Estw akolouja (an) kai [0, 1). Upojtoume ti upr-qei k N ste |an+1| |an| gia kje n k. Tte to rio limn an uprqeikai enai 0.

(ii) An an 6= 0 gia kje n N kai h akolouja ( |an+1||an| ) sugklnei snanarijm < 1, tte to rio limn an uprqei kai enai 0.

Apdeixh Jtw bn = |an|.(i) Prpei na dexw ti to limn bn uprqei kai enai 0. 'Eqoume bk+1 bk, bk+2 bk+1 bk2 kai epagwgik

0 bk+n bkn gia kje n N.

All efson 0 < < 1 qoume n 0 kai ra limn bn = limn bn+k = 0.

'Allh apdeixh: Afo bn+1 bn bn gia kje n k, h (bn) enai telikfjnousa epomnwc sugklnei, stw sto b 0. 'Omwc, afo bn+1 bn giakje n k qoume

0 b = limnbn+1 lim

nbn = b

ra b(1 ) 0. Efson b 0 kai 1 > 0, petai ti b = 0.

(ii) 'Estw ti h akolouja ( |an+1||an| ) sugklnei snan arijm < 1. Epilgw

ste < < 1 (uprqei ttoio giat < 1). Afo bn+1bn

uprqei kste bn+1

bn< gia kje n k. 'Epetai ap to (i) ti to rio limn bn uprqei

kai enai 0. 2

Paratrhsh 3.20 Kai ed oi upojseic 0 < 1, 0 < 1 den mpo-ron en gnei na paraleifjon. Gia pardeigma an an = 1+

1ntte an+1 < an

gia kje n all h (an) den sugklnei sto 0.

Prtash 3.21 An 0 < a < 1 kai k Z tte limn nkan = 0.

Apdeixh Jtontac xn = nkan qoume

|xn+1||xn|

=(n+ 1)kan+1

nkan=

(1 +

1

n

)ka a .

Efson a < 1, to apotlesma petai ap to Lmma. 2

11

Prtash 3.22 H akolouja (an) pou an = (1+1n)n sugklnei snan arijm

e me 2 < e < 3.

Prth apdeixh. Isqurismc 1 H (an) enai axousa, dhl. an+1 angia kje n:

Qrhsimopoi thn anisthta arijmhtiko msou - gewmetriko msou

x1 + x2 + . . .+ xn+1n+ 1

n+1x1x2 . . . xn+1

gia touc jetikoc arijmoc

x1 = x2 = . . . = xn = 1 +1

nkai xn+1 = 1

opte qw

n(1 + 1n) + 1

n+ 1 n+1

(1 +

1

n

)n.1

dhlad

1 +1

n+ 1 n+1

(1 +

1

n

)n.

Uynontac kai ta do mlh thc anisthtac autc (pou enai jetik) sthn n+1,prokptei h epijumht anisthta(

1 +1

n+ 1

)n+1(1 +

1

n

)n.

Isqurismc 2 H (an) enai fragmnh, kai mlista 2 < an < 3 gia kjen 2:'Eqoume a1 = 2, a2 =

(32

)2= 9

4> 2 kai gia kje n 2, ap to diwnumik

anptugma

an = 1 +n

k=1

n(n 1)(n 2) . . . (n k + 1)k!

1

nk

= 1 +n

k=1

1

k!

n

n(1 1

n)(1 2

n) . . . (1 k 1

n) < 1 +

nk=1

1

k!.

12

All k! = 1.2.3 . . . k, ra, tan k 2, tte k! 1.2.2 . . . 2 = 2k1 opte

an < 1 +n

k=1

1

k! 1 +

nk=1

1

2k1.

'Omwc pwc enai gnwst to jroisma twn rwn gewmetrikc prodou enain

k=1

1

2k1= 1 +

1

2+

1

22+ . . .+

1

2n1=

1 12n

1 12

= 2 12n1

< 2

kai sunepc an < 3 gia kje n.

'Epetai tra ap to Jerhma 3.12 ti h (an) sugklnei, kai an jsoume e =lim an tte efson

94 an < 3 gia kje n 2 qoume 2 < 94 e 3, all

den mporome amswc na sumpernoume ti e < 3.

Mporome mwc na beltisoume thn prosggish wc exc: an n 4,

an < 1 +n

k=1

1

k!= 1 +

(1 +

1

2+

1

2.3+

nk=4

1

k!

) 2 + 1

2+

1

6+

nk=4

1

2k1

2 + 12+

1

6+

123 1

2n

1 12

< 2 +1

2+

1

6+

1

4= 2 +

11

12

ra e 2 + 1112< 3.

Deterh apdeixh 'Estw bn = (1 +1n)n+1.

Isqurismc H (an) enai axousa kai h (bn) enai fjnousa.

Apdeixh Isqurismo: Gia n 2, qoume an = (1 + 1n)n =

(n+1n

)n, bn1 =

(1 + 1n1)

n =(

nn1

)n, opte

anbn1

=(1 + 1

n)n

(1 + 1n1)

n=

(n+ 1

n

)n(n 1n

)n=

(n2 1n2

)n=

(1 1

n2

)nBer> 1 n 1

n2= 1 1

n=

n 1n

pou qrhsimopohsa thn anisthta Bernoulli. Epomnwc

an > bn1

(n 1n

)=

(n

n 1

)n(n 1n

)=

(n

n 1

)n1=

(1 +

1

n 1

)n1= an1.

13

ra h (an) enai axousa. Epshc

bn1an

=

(n2

n2 1

)n=

(1 +

1

n2 1

)n>

(1 +

1

n2

)nBer> 1+n

1

n2= 1+

1

n=

n+ 1

n

opte

bn1 > an

(n+ 1

n

)=

(n+ 1

n

)n(n+ 1

n

)=

(n+ 1

n

)n+1= bn

ra h (bn) enai fjnousa. Endeiktik, oi prtoi roi enai:

a1 = (1 +11)1 = 2 b1 = (1 +

11)2 = 4

a2 = (1 +12)2 = 9

4= 2.25 b2 = (1 +

12)3 = 27

8= 3.375

a3 = (1 +13)3 = 64

27' 2.3704 b3 = (1 + 13)

4 = 25681

' 3.1605a4 = (1 +

14)4 = 625

256' 2.4414 b4 = (1 + 14)

5 = 31251024

' 3.0518a5 = (1 +

15)5 = 7776

3125' 2.4883 b5 = (1 + 15)

6 = 4665615625

' 2.986

'Epetai tra ap to Jerhma 3.12 ti kai oi do akoloujec sugklnoun.Mlista epeid to phlko bn

an= 1 + 1

nsugklnei sto 1, oi do akoloujec ja

qoun to dio rio, e.Tloc, gia n > 5,

9

4= a2 < an < bn < b5 =

(6

5

)6=

46656

15625< 3

ra 2 < an e bn < 3 . 2

Paratrhsh 3.23 Oi anisthtec(1 +

1

n

)n< e 0. Uprqei no Nste |an| < 2 tan n no. 'Estw n > no. 'Eqoume

|bn| =a1 + a2 + . . .+ anon + ano+1 + ano+2 + . . .+ ann

|a1|+ |a2|+ . . .+ |ano |

n+|ano+1|+ |ano+2|+ . . .+ |an|

n

no ste giakje n n1 na qoume

|a1|+ |a2|+ . . .+ |ano |n

kn ste akn+1 > akn .'Etsi kataskeusame mia (gnhswc) axousa upakolouja (akn) thc (an).2

Apdeixh tou Jewrmatoc 4.4 'Estw (an) mia fragmnh akolouja.Ap thn teleutaa Prtash, h (an) qei mia montonh upakolouja, stw (akn),h opoa, wc upakolouja fragmnhc akoloujac, enai kai aut fragmnh1.Dhlad h (akn) enai montonh kai fragmnh, ra (Jerhma 4.12) sugklnei.2

1an |an| M gia kje n, tte bebawc |akn | M gia kje n

2

Deterh apdeixh tou Jewrmatoc 4.4 Afo h (an) enai fragmnh,uprqoun x0, y0 R ste x0 an y0 gia kje n N, dhlad an Ipou I = [x0, y0] disthma mkouc = y0 x0. Diairome to disthma Ise do sa upodiastmata mkouc /2. Toulqiston na ap aut periqeita an gia peiro pljoc deiktn n. 'Estw I1 na upodisthma pou periqeipeirouc rouc thc (an); an kai ta do upodiastmata periqoun peiroucrouc, onomzoume I1 aut me to mikrtero ktw kro. Epilgoume ak1 I1.Diairome tra to disthma I1 se do sa upodiastmata mkouc /22. Plina ap aut, stw to I2, periqei peirouc rouc thc (an) (pli epilgoumeto {aristertero}, an kai ta do periqoun peirouc rouc). Epeid to I2periqei peirouc rouc thc (an), periqei kai kpoion ro ak2 me k2 > k1.Suneqzoume me autn ton trpo. An qoume kataskeusei In In1 pou naperiqei peirouc rouc thc (an), epilgoume na kn > kn1 ste akn In,diairome to In se do sa upodiastmata, na ap aut ja periqei peiroucrouc thc (an), to onomzoume In+1.

Sqhmatzoume tsi mia akolouja diasthmtwn In = [xn, yn] me In+1 Ingia kje n kai mia upakolouja (akn) thc (an) me akn In gia kje n. EfsonIn+1 In, qoume xn xn+1 < yn+1 yn, opte h (xn) enai axousa kai h(yn) enai fjnousa. Epshc h (xn) enai nw fragmnh (ap kje ro thc (yn))kai h (yn) enai ktw fragmnh (ap kje ro thc (xn)). Ap to Jerhma 4.12petai ti kai oi do ja sugklnoun, stw limn xn = x kai limn yn = y. Afoxn yn gia kje n qoume x y. All epiplon qoume yn xn = 2n 0,epomnwc x = y. Afo akn In, dhlad xn akn yn gia kje n, petaitra ti kai h (akn) ja sugklnei (sto x).

2 2

Paratrhsh Shmeinoume ti krsimo rlo kai stic do apodexeic paixeh plhrthta tou R, msw tou Jewrmatoc 4.12.

Paratrhsh 4.6 An (an) enai mia sugklnousa akolouja, tte diaisjh-tik oi roi thc plhsizoun aujareta kont sto ri thc, ra plhsizounaujareta kont metax touc. Akribstera:

An an a, tte dojntoc > 0 uprqei no N me |an a| < /2 giakje n no. Sunepc an oi dektec m kai n enai kai oi do megalteroi soi tou no, tte

|am an| |am a|+ |a an| < .2Apodexame ed mia eidik perptwsh thc arqc tou kibwtismo diasthmtwn: an J1

J2 . . . Jn . . . enai mia akolouja kleistn diasthmtwn pou ta mkh touc tenounsto 0, tte uprqei na kai monadik shmeo pou periqetai se la ta Jn. Apdeixh: skhsh.

3

H teleutaa aut idithta odhge ston orism:

Orismc 4.1 Mia akolouja (an) lgetai basik akolouja Cauchyan

gia kje > 0 uprqei no N (pou exarttai ap to )ste gia kje m kai n me m,n no na isqei |am an| < .

Paratrhsh 4.7 Gia na enai mia akolouja basik, prpei la ta zeu-gria {an, am} na plhsizoun telik metax touc. Den arke p.q. na isqeian an+1 0.

Pardeigma 4.8 An an = 1 +12

+ . . . + 1nqw an+1 an = 1n+1 0 all

|a2n an| =(

1 + . . . +1

n+

1

n + 1+ . . . +

1

2n

)

(1 + . . . +

1

n

)=

1

n + 1+

1

n + 2+ . . . +

1

2n n 1

2n=

1

2,

(diti 1n+k

12n

gia k = 1, . . . , n) sunepc h (an) den enai basik3.

Sthn Paratrhsh 4.6 dexame ti mia sugklnousa akolouja enai pntabasik. Exaitac thc plhrthtac (!) tou sunlou twn pragmatikn arijmn,isqei kai to antstrofo:

Jerhma 4.9 Mia akolouja sugklnei an kai mnon an enai basik.

Apdeixh Prpei na dexoume ti kje basik akolouja sugklnei. Jadexoume ti:

1. Kje basik akolouja enai fragmnh.

'Epetai tte ap to Jerhma Bolzano - Weierstrass ti kje basik ako-louja qei mia upakolouja pou sugklnei. H apdeixh oloklhrnetaiapodeiknontac ti:

2. An mia basik akolouja (an) qei mia upakolouja pou sugklnei sekpoio a R, tte h dia h (an) sugklnei sto a.

3Parathrome mlista ti kje for pou o dekthc n tou an diplasizetai, to an auxneikat 12 , epomnwc h an enai axousa kai qi nw fragmnh, ra an + (apdeixh:skhsh).

4

Lmma 4.10 Kje basik akolouja enai fragmnh.

Apdeixh 4 'Estw ti h (an) enai basik. Tte gia kje > 0 uprqeino N ste an m,n no na isqei |an am| < . Epomnwc, gia kjen no, qoume |an ano | < , ra |an| < |ano |+ . Dhlad to snolo

{ano , ano+1, ano+2, . . .} = {an : n no}

enai fragmno. 'Omwc to {uploipo} snolo

{a1, a2, . . . , ano1}

enai peperasmno, sunepc kai aut fragmno. 'Ara kai h nwsh twn dosunlwn, dhlad to {an : n N}, ja enai fragmno. 2

Lmma 4.11 An mia basik akolouja (an) qei mia upakolouja (akn) pousugklnei se kpoio a, tte h dia h (an) sugklnei sto a.

Apdeixh 'Estw > 0.Afo h (an) enai basik,

uprqei no N ste gia kje m,n no na isqei |am an| 0 kai an+1 = 1+2

1 + an=

an + 3an + 1

. H (an) sugklnei, mlista sto

3.

Apdeixh Paratrhsh: an > 0 gia kje n (epagwg). 'Ara h (an) enai kalorismnh (an + 1 6= 0 gia kje n). Epshc (afo an > 0)

0 < an+1 =an + 3an + 1

0 afo an > 0. Omowc

a2n+1 a2n1 =2

1 + a2n 2

1 + a2n2= yn(a2n a2n2)

pou yn = 2(1+a2n)(1+a2n2) > 0 afo an > 0. 'Eqoume loipn

a2n+2 a2n = xn(a2n+1 a2n1) = +xnyn(a2n a2n2)

'Ara an bn = a2n+2a2n qw bn = xnynbn1. 'Epetai ti h (a2n) enai montonh:an a4 a2, dhlad b1 0, tte (epagwgik) bn 0 gia kje n, dhlad a2n+2 a2n gia kje n, dhlad (a2n) axousa. Kai an a4 a2 tte bn 0 gia kje n,opte h (a2n) enai fjnousa.

Ap th sqsha2n+1 a2n1 = yn(a2n a2n2)

prokptei tra ti kai h (a2n+1) enai montonh: an h (a2n) enai axousa, dhl.a2na2n2 0 gia kje n tte a2n+1a2n1 0 gia kje n, dhlad h (a2n+1)enai fjnousa, kai omowc an h (a2n) enai fjnousa tte h (a2n+1) enai axousa.

Sunepc kai oi do enai montonec kai fragmnec, ra sugklnoun. Mnei nadeiqje ti qoun to dio rio. An a2n x kai a2n1 y qoume

x = lim a2n = lima2n1 + 3a2n1 + 1

=y + 3y + 1

kai y = lim a2n+1 = lima2n + 3a2n + 1

=x + 3x + 1

opte

x =y + 3y + 1

=

x + 3x + 1

+ 3

x + 3x + 1

+ 1=

2x + 3x + 2

kai omowc

y =x + 3x + 1

=2y + 3y + 2

ap tic opoec brskoume x2 + 2x = 2x + 3 kai y2 + 2y = 2y + 3 ra x2 = y2 = 3opte (epeid x 0 kai y 0) x = y =

3.

1

5 Sunartseic

An X, Y enai do mh ken snola, mia sunrthsh ap to X sto Y enai miaantistoqish, pou antistoiqzei se kje stoiqeo tou X, na kai mona-dik stoiqeo tou Y . Den uprqei kannac periorismc sthn antistoqishaut. Arke kje stoiqeo tou X na qei ma kai monadik {eikna} sto Y .

Dhlad sunrthsh enai na snolo f ap diatetagmna zegh1 (x, y) meprto stoiqeo x X kai detero stoiqeo y Y , pou ikanopoie tic excdo proupojseic:

(a) Gia kje x X uprqei y Y ste (x, y) f kai(b) Aut to y enai monadik, dhlad an (x, y1) f kai (x, y2) ftte y1 = y2.

An x X, to monadik y Y gia to opoo isqei (x, y) f sumbolzoumef(x) kai grfoume sunjwc f : X Y gia na dhlsoume ti h f enai miasunrthsh ap to X sto Y .

An f : X Y enai mia sunrthsh, to X onomzetai pedo orismo kaito Y pedo timn thc f . Sthn pragmatikthta mia sunrthsh apoteletaiap tra antikemena: (i) to pedo orismo X (ap pou {xekinei} h sunr-thsh), (ii) to pedo timn Y (pou {katalgei}) kai (iii) thn {antistoqish}enc kai monadiko shmeou f(x) tou Y se kje shmeo x tou X.

Snolo timn eikna thc f enai to snolo

f(X) = {y Y : uprqei x X ste f(x) = y} {f(x) : x X} Y.

H sunrthsh f lgetai ep an f(X) = Y , dhlad an gia kje y Yuprqei kpoio (endeqomnwc poll) x X ste f(x) = y.

H sunrthsh f lgetai 1-1 an kje y f(X) enai h eikna monadikox X, dhlad an gia kje x1, x2 X me x1 6= x2 qoume f(x1) 6= f(x2), isodnama, an ap tic sqseic (x1, y) f kai (x2, y) f petai ti x1 = x2.

An f : X Y kai g : W Z enai sunartseic, kai an f(X) W , tteh snjesh g f : X Z orzetai wc exc

Xf f(X) W g Z

x f(x) g(f(x))1dhlad na uposnolo tou kartesiano ginomnou X Y

1

dhlad(g f)(x) = g(f(x)) (x X).

An X enai tuqao snolo, h tautotik sunrthsh idX : X X orzetaiwc exc: idX(x) = x, x X.

Prtash 5.1 'Estw f : X Y mia sunrthsh. TteUprqei g : Y X ste g f = idX an kai mnon an h f enai 1-1.

Gia pardeigma an f : R+ R : x x2 mpor na orsw g : R R+ ap

ton tpo g(y) =

{ y an y 0

16 an y < 0(bebawc h epilog tou 16 enai aujareth).

'Eqw tte gia kje x R+, g(f(x)) = g(x2) =x2 = x, afo x 0.

Paratrhse mwc ti h g den ikanopoie th sqsh f g = idR. Prgmati,an y < 0 tte f(g(y)) = f(16) = 256 6= y.

Epshc, den uprqei mnon ma, all pollc sunartseic g pou ikanopoionth sqsh g f = idR+ .

Aut sumbanei epeid h f den enai ep.

'Estw f : X Y ma sunrthsh. 'Opwc enai gnwst, uprqei ma sunr-thsh g : Y X pou ikanopoie kai tic do sqseic f g = idY kai g f = idXan kai mnon an h f enai 1-1 kai ep. H sunrthsh aut, tan uprqei, enaimonadik. Onomzetai h antstrofh thc f kai sumbolzetai f1.

Kataqrhstik ja qrhsimopoiome to smbolo f1 tan h f enai 1-1, giathn antstrofh thc sunrthshc f : X f(X):

Orismc 5.1 An h f enai 1-1, tte h antstrofh sunrthsh f1

orzetai sto snolo timn f(X) wc exc:

f1 : f(X) X : f(x) x

Dhlad gia kje y f(X), h tim f1(y) enai o monadikc arijmc x Xpou ikanopoie th sqsh f(x) = y.

Prtash 5.2 An h f : X Y enai 1-1, tte orzetai h f1 : f(X) Xkai

(i) (f1 f)(x) = x gia kje x X (dhlad f1 f = idX) kai(ii) (f f1)(y) = y gia kje y f(X) (dhlad f f1 = idf(X)).

2

'Estw f : X Y mia sunrthsh.Se kje A X antistoiqome thn eikna tou A sto Y , pou enai to

snolo

f(A) {y Y : uprqei x A ste f(x) = y} {f(x) Y : x A}.

Se kje B Y antistoiqome thn antstrofh eikna tou B sto X,pou enai to snolo

f1(B) {x X : f(x) B}

(Shmewse ti to f1(B) enai na uposnolo touX, den enai en gnei h eiknatou B msw kpoiac sunrthshc2.)

Prtash 5.3 'Estw f : X Y mia sunrthsh.

(i) An A1 A2 X tte f(A1) f(A2).

(ii) An A1, A2 enai uposnola tou X tte f(A1 A2) = f(A1) f(A2).

(iii) An A1, A2 enai uposnola tou X tte f(A1 A2) f(A1) f(A2).Isthta isqei tan h f enai 1-1, all qi en gnei.

(iv) An B1 B2 Y tte f1(B1) f1(B2).

(v) An B1, B2 enai uposnola tou Y tte f1(B1B2) = f1(B1)f1(B2).

(vi) An B1, B2 enai uposnola tou Y tte f1(B1B2) = f1(B1)f1(B2).

(vii) An B Y tte f1(Y \B) = X \ f1(B).

(viii) An A X tte A f1(f(A)). Isthta isqei tan h f enai 1-1, allqi en gnei.

(ix) An B Y tte f(f1(B)) B. Isthta isqei tan h f enai ep, allqi en gnei.

Paradegmata 5.4 Gia to (iii), pre p.q. f : R R me f(x) = x2 kai A1 =[1, 0], A2 = [0, 2].Gia to (viii), pre A = [12 , 1] opte f(A) = [0, 1] kai f

1(f(A)) = [1, 1] 6= A.Gia to (ix), an B = [4, 1] tte f1(B) = [1, 1] ra f(f1(B)) = [0, 1] 6= B.

2An bbaia h f enai 1-1 kai B f(X), tte to f1(B) enai h eikna tou B msw thcsunrthshc f1

3

Orismc 5.2 An f : X R kai g : X R enai sunartseic tte orzoumetic sunartseic

f + g : X R pou (f + g)(x) = f(x) + g(x) gia kje x R.

f g : X R pou (f g)(x) = f(x) g(x) gia kje x R.

Eidiktera an R orzoume thn sunrthsh f : X R ap th sqsh(f)(x) = f(x) gia kje x R.

An g(x) 6= 0 gia kje x X orzoume thn fg

: X R ap th sqsh(f

g

)(x) =

f(x)

g(x)gia kje x R.

An c R, h stajer sunrthsh c : X R orzetai ap th sqshc(x) = c gia kje x R.

Eidiktera h mhdenik sunrthsh 0 : X R orzetai ap thsqsh 0(x) = 0 gia kje x R.

An f : X R kai g : X R enai do sunartseic, ja lme ti

f g an f(x) g(x) gia kje x X.

Parathrome ti h sqsh aut enai sqsh merikc (kai qi olikc) ditaxhc.Dhlad mpore na mhn isqei ote h f g ote h g f . Gia pardeigma anf, g : R R me f(x) = x kai g(x) = x3 qoume f(1

2) g(1

2) en f(2) g(2).

Orismc 5.3 'Estw X R kai f : X R mia sunrthsh.

(i) H f lgetai axousa an: x, y X, x < y f(x) f(y).

(ii) H f lgetai gnhswc axousa an: x, y X, x < y f(x) < f(y).

(iii) H f lgetai fjnousa an: x, y X, x < y f(x) f(y).

(iv) H f lgetai gnhswc fjnousa an: x, y X, x < y f(x) > f(y).

(v) H f lgetai montonh an enai axousa fjnousa.

(vi) H f lgetai gnhswc montonh an enai gnhswc axousa gnhswcfjnousa.

4

(vii) H f lgetai nw fragmnh an to f(X) enai nw fragmno snolo,an dhlad uprqei M R ste gia kje x X na isqei f(x) M .

(viii) H f lgetai ktw fragmnh an to f(X) enai ktw fragmno snolo,an dhlad uprqei L R ste gia kje x X na isqei f(x) L.

(ix) H f lgetai fragmnh an enai nw kai ktw fragmnh, isodnama anuprqei K R ste gia kje x X na isqei |f(x)| K.

Prtash 5.5 'Estw X R kai f : X R gnhswc montonh sunrthsh.Tte(i) H antstrofh sunrthsh f1 : f(X) R uprqei.(ii) H f1 enai gnhswc axousa an kai mnon an h f enai gnhswc axousa.(iii) f1 enai gnhswc fjnousa an kai mnon an h f enai gnhswc fjnousa.

5

6 Suneqec Sunartseic

'Estw X, Y uposnola tou R kai f : X Y mia sunrthsh. H f enaisuneqc sna shmeo xo tou pedou orismo thc an h tim f(xo) mpore naproseggisje me aujareth akrbeia an prei kanec timc f(x) se shmea xarket geitonik sto xo. O akribc orismc enai o exc:

Orismc 6.1 'Estw X, Y uposnola tou R kai f : X Y mia sunrthsh.H f lgetai suneqc sna shmeo xo X an gia kje (aujareto) > 0uprqei (katllhlo) > 0 ste:

an x X kai |x xo| < tte |f(x) f(xo)| < .H f lgetai suneqc sto X an enai suneqc se kje xo X.

Parathrseic 6.1 (a) H sunqeia thc f sto xo mpore na diatupwje kaiwc exc: An doje opoiadpote znh

Z = (f(xo) , f(xo) + )

grw ap thn tim f(xo) mporome na brome mia geitoni

= (xo , xo + ) X

tou xo pou na apeikonzetai olklhrh msa sth znh Z, dhlad ttoia stef() Z.

(b) H sunqeia qei nnoia mnon se shmea xo tou pedou orismo thc f(bl. p.q. to Pardeigma 6.5).

(g) O arijmc exarttai en gnei kai ap to kai ap to xo.

(d) An uprqei na > 0 pou ikanopoie ton orism tte kje me 0 < < epshc ton ikanopoie.

Efarmzontac ton orism mporec na apodexeic ti oi akloujec sunartseicenai suneqec:

Pardeigma 6.2 f : R R me f(x) = 3 gia kje x.

Enai suneqc se kje xo R. (Ed to den exarttai ote ap to oteap to xo.)

Pardeigma 6.3 f : R R me f(x) = x gia kje x.

1

Enai suneqc se kje xo R. (Ed to exarttai ap to all qi ap toxo: kje me 0 < ikanopoie ton orism.)

Pardeigma 6.4 f : R R me f(x) = 5x2 + 6 gia kje x.

Enai suneqc se kje xo R.Apdeixh An doje > 0, jloume na prosdiorsoume na katllhlo > 0ste na ikanopoietai o orismc. An |x xo| < tte

|f(x) f(xo)| = 5|x2 x2o| = 5|x xo|.|x+ xo| < 5|x+ xo|.

Arke loipn na brome > 0 ste 5|x + xo| < gia la ta x sth geitoni = (xo , xo + ) tou xo. Den blptei th genikthta na periorsoumekatarqn to x se mia geitoni pltouc p.q. 1 ( gia na dieukolnoume tic pr-xeic) kai argtera mporome na to periorsoume akma perisstero (anlogame to ). Aut shmanei ti anazhtome metax 0 kai 1. Parathrome ti an|x xo| < 1 tte |x| < |xo| + 1 opte |x + xo| |x| + |xo| < 2|xo| + 1. Anloipn dialxoume na (0, 1] pou na ikanopoie

5(2|xo|+ 1)tte gia

kje x me |x x0| < ja qoume 5|x+ xo| < 5(2|xo|+ 1), ra

|f(x) f(xo)| = 5|x xo|.|x+ xo| 5(2|xo|+ 1) < .

Dhlad sto sugkekrimno pardeigma o orismc thc sunqeiac ikanopoietai

gia kje me 0 < min{1, 5(2|xo|+ 1)

}. Ed to pou brkame1 exarttai

kai ap to kai ap to xo.

Pardeigma 6.5 f : R\{0} R me f(x) = 1xgia kje x 6= 0.

Enai suneqc se kje xo tou pedou orismo thc.Apdeixh Parathrome prta ti, efson xo 6= 0, an epilxoume to x ste|x xo| < |xo|2 , tte |x| >

|xo|2

(trigwnik anisthta), opte (bebawc x 6= 0kai) 1x 1xo

= |x xo||x|.|xo| < 2 |x xo|x2o .1Mpore na apodeiqje (pwc ja dome se epmeno keflaio) ti den uprqei sto par-

deigma aut > 0 pou na mhn exarttai kai ap to xo.

2

Epomnwc an doje > 0, epilgontac me 0 < min{ |xo|2, x

2o

2} ja qoume,

gia kje x me |xxo| < (opte to x ja ankei automtwc sto pedo orismothc f) ti

|f(x) f(xo)| =1x 1xo

< 2 |x xo|x2o < 2 x2o . 2Mporome na epektenoume th sunrthsh aut slo to R ste na enai

panto suneqc? 'Oqi:

Pardeigma 6.6 fc : R R me fc(x) =

1x, x 6= 0

c, x = 0.

H fc den enai suneqc sto 0, poia ki an enai h tim tou c.

Pardeigma 6.7 g : R R me g(x) =

sin 1

x, x 6= 0

0, x = 0.

H g den enai suneqc sto 0.(Ta do aut paradegmata apodeiknontai eukoltera qrhsimopointac

thn Prtash 6.9.)

Pardeigma 6.8 h : R R me h(x) =

x sin 1

x, x 6= 0

0, x = 0.

H h enai suneqc sto 0.Apdeixh 'Estw > 0. Uprqei tte > 0 (mlista, mporome na proume = ) ste gia kje x me |x 0| < na isqei

|h(x) h(0)| =x sin 1x

|x| < (efson | sin | 1 gia kje R).

Prtash 6.9 Ma sunrthsh f : X Y enai suneqc sna shmeo xo X an kai mnon an gia kje akolouja (xn) pou sugklnei sto xo, me xn Xgia kje n, h akolouja (f(xn)) sugklnei sto f(xo).

3

Apdeixh Upojtoume ti h f enai suneqc sto xo. 'Estw (xn) akoloujasto X me xn xo. Ja dexw ti f(xn) f(xo). 'Estw > 0. Jlw nadexw ti uprqei no N ste |f(xn) f(xo)| < gia kje n no. Afoh f enai suneqc sto xo, uprqei > 0 ste2 f() Z. All xn xo,ra uprqei no N ste |xn xo| < gia kje n no. Aut shmanei tigia kje n no qoume xn (efson xn X ex upojsewc) kai raf(xn) Z, dhlad |f(xn) f(xo)| < .

Gia to antstrofo, upojtoume ti h f den enai suneqc sto xo. Uprqeitte kpoio > 0 ste (gia kje > 0, ra kai) gia kje n thc morfcn =

1nna mhn isqei ti f(n) Z , opte mporome na brome xn n

me f(xn) / Z. Dhlad qoume xn X kai |xn xo| < 1n gia kje n, opte hakolouja (xn) sugklnei sto xo, all |f(xn)f(xo)| , opte h akolouja(f(xn)) den sugklnei sto f(xo). 2

Tonzoume ti gia na dexoume ti h f enai suneqc sto xo prpei nadexoume ti f(xn) f(xo) gia opoiadpote akolouja (xn) apo to X pousugklnei sto xo. Gia na dexoume ti h f den enai suneqc sto xo arke nabrome mia akolouja (xn) apo to X pou na sugklnei sto xo, all h (f(xn))na mhn sugklnei sto f(xo) (na sugklnei allo, poujen).

Apodexeic twn paradeigmtwn 6.6 kai 6.7 Jewrome thn akolouja

(xn) =

(1

n+ 2

). Oi roi thc brskontai loi sto pedo orismo twn sunar-

tsewn, kai h akolouja tenei sto 0, all h akolouja (fc(xn)) = (n+

2)

den sugklnei sto c kai h akolouja (g(xn)) = ((1)n) den sugklnei sto 0.

Prtash 6.10 'Estw X R kai xo X. An oi sunartseic f, g : X Renai suneqec sto xo, tte

1. H sunrthsh f + g enai suneqc sto xo.

2. Gia kje R, h sunrthsh f enai suneqc sto xo.

3. H sunrthsh f.g enai suneqc sto xo.

4. An ep plon g(x) 6= 0 gia kje x X, tte h sunrthsh fgenai (kal

orismnh sto X kai) suneqc sto xo.

2Qrhsimopoi touc sumbolismoc thc Paratrhshc 6.1.

4

H apdeixh enai mesh sunpeia twn antistoqwn idiottwn twn orwn ako-loujin.

Gia pardeigma, gia na dexw ti hf

genai suneqc sto xo, arke na dexw

ti, gia kje akolouja (xn) pou sugklnei sto xo, me xn X gia kje n, h

akolouja

(f

g(xn)

)sugklnei sto

f

g(xo). All afo oi f kai g enai suneqec

sto xo, qoume f(xn) f(xo) kai g(xn) g(xo) ap thn Prtash 6.9.

Epiplon isqei ti g(xn) 6= 0 gia kje n. Kat sunpeia to phlko(f(xn)

g(xn)

)twn akoloujin (orzetai kai) sugklnei sto phlko

f(xo)

g(xo)twn orwn touc.

Pardeigma 6.11 H sunrthsh Dirichlet f(x) =

1 x Q

0 x / Qenai asuneqc se kje xo R.

Prgmati, xroume ti (poio kai na enai to xo) uprqei akolouja (xn) mexn Q gia kje n ste xn xo, kai uprqei akolouja (yn) me yn / Q giakje n ste yn xo. 'Eqoume f(xn) = 1 gia kje n, ra f(xn) 1, enf(yn) = 0 gia kje n opte f(yn) 0. Epomnwc h f den mpore na enaisuneqc sto xo, giat an tan ja prepe na isqoun oi isthtec limn f(xn) =f(xo) = limn f(yn), en isqei limn f(xn) 6= limn f(yn).

Prtash 6.12 O periorismc mic suneqoc sunrthshc enai suneqc: 'E-stw xo Y X R kai f : X R mia sunrthsh. An h f enai suneqcsto xo, tte o periorismc thc sto Y , stw g = f |Y , enai suneqc.

To antstrofo den isqei pnta: Mia asuneqc sunrthsh enai dunatnna qei suneq periorism.

Apdeixh An doje > 0 prpei na brome > 0 stex Y (xo , xo + ) |g(x) g(xo)| < .

All h f enai suneqc sto xo, ra uprqei > 0 stex X (xo , xo + ) |f(x) f(xo)| < .

An mwc x Y (xo , xo + ) tte x X (xo , xo + ) kai ra|g(x) g(xo)| = |f(x) f(xo)| < .

'Ena pardeigma asuneqoc sunrthshc me suneq periorism enai h su-nrthsh Dirichlet, stw f . 'Estv xo Q. O periorismc f |Q enai h stajersunrthsh 1, ra h f |Q enai suneqc. 'Omwc h f enai asuneqc sto xo. 2

5

Enai faner ap ton orism thc sunqeiac ti h sumperifor mic sunr-thshc f {makri} ap to xo den ephrezei thn sunqei thc sto xo: aut pouendiafrei enai h sumperifor thc {topik}, se mia (osodpote mikr) geitonitou xo. Prgmati:

Prtash 6.13 (Topik idithta) 'Estw xo X R kai f : X Rmia sunrthsh. An uprqei > 0 ste o periorismc thc f sto X (xo , xo + ) na enai suneqc sto xo, tte h dia h f enai suneqc sto xo.

Apdeixh An = X (xo , xo + ) kai g = f | tte, ap thn upjesh,gia kje > 0 uprqei 1 > 0 ste gia kje x (xo 1, xo + 1) naisqei |g(x) g(xo)| < .

'Estw = min{, 1}. An x X (xo, xo+) tte x (xo, xo+)kai ra x . Epshc x (xo1, xo+1) ra |g(x)g(xo)| < . All afoh g enai o periorismc thc f sto qoume |f(x)f(xo)| = |g(x)g(xo)| < .Deterh apdeixh Ap thn Prtash 6.9, an (xn) enai tuqaa akolouja stoi-qewn tou X pou sugklnei sto xo, prpei na dexoume ti h akolouja (f(xn))sugklnei sto f(xo). Afo xn xo, uprqei no N ste xn (xo, xo+)gia kje n no. Epomnwc gia kje n no to xn ankei sto pedo orismo thc g, h opoa enai suneqc sto xo. 'Epetai ap thn Prtash 6.9 ti hakolouja (g(xn))nno sugklnei sto g(xo). All g(xn) = f(xn) tan n nokai g(xo) = f(xo). Epomnwc kai h (f(xn)) sugklnei sto f(xo) (afo h(f(xn))nno sugklnei sto f(xo)). 2

Prisma 6.14 (1) Kje poluwnumik sunrthsh p enai suneqc se kjexo R.

(2) Kje rht sunrthsh f = pq(pou p, q enai polunuma) enai suneqc

sto pedo orismo thc X = {x R : q(x) 6= 0}.

Mia suneqc sunrthsh f den enai katangkhn fragmnh slo to pedoorismo thc. Jerhse gia pardeigma tic sunartseic f(x) = x2 g(x) = 1

x.

Mporome mwc, grw ap kje shmeo tou pedou orismo, na brome mia(endeqomnwc mikr) perioq sthn opoa o periorismc thc f enai fragmnhsunrthsh:

Prtash 6.15 'Estw xo X R kai f : X R. An h f enai suneqcsto xo, tte uprqei > 0 kai M > 0 ste gia kje x X (xo , xo + )na isqei |f(x)| M).

6

Apdeixh An doje > 0, ap ton orism thc sunqeiac sto xo uprqei > 0 ste gia kje x X (xo , xo + ) na isqei |f(x) f(xo)| < opte |f(x)| < |f(xo)| + . Dhlad o arijmc M = |f(xo)| + apotele nafrgma tou f(). 2

'Otan isqei to sumprasma thc Prtashc, tan dhlad uprqei > 0ste to snolo f() na enai fragmno, lme ti h f enai topik fragmnhse mia perioq tou xo.

Prtash 6.16 'Estw xo X R kai f : X R suneqc sto xo. An f(xo) > 0 uprqei perioq = X (xo , xo + ) tou xo stef(x) > 0 gia kje x .

An f(xo) < 0 uprqei perioq tou xo ste f(x) < 0 gia kje x .

Apdeixh Jtoume = 13|f(xo)|: enai jetikc arijmc. Ap thn sunqeia

thc f sto xo uprqei > 0 ste gia kje x na isqei

13|f(xo)| < f(x) f(xo) 0 h anisthta f(x)f(xo) > 13f(xo) dnei f(x) >23f(xo) > 0 gia

kje x en an f(xo) < 0 h anisthta f(x)f(xo) < 13 |f(xo)| = 13f(xo)

dnei f(x) < 23f(xo) < 0 gia kje x . 2

Paratrhsh 6.17 H epilog tou = 13|f(xo)| enai aujareth: opoiad-

pote jetik tim, gnsia mikrterh ap |f(xo)|, ja arkose gia thn apdeixh.Me thn epilog aut dexame ti an f(xo) > 0 uprqei perioq tou xo

ste gia kje x na isqei (qi aplc f(x) > 0 all) f(x) > 23f(xo).Epilgontac p.q. = 1

10|f(xo)| mporome na brome perioq ste

f(x) > 910f(xo) gia kje x .

Prtash 6.18 'Estw xo X R kai f : X R mia sunrthsh. An h fenai suneqc (sto xo) tte h sunrthsh |f | enai suneqc (sto xo).

Apdeixh Jtoume g = |f |. Gia kje > 0 uprqei > 0 ste gia kjex X (xo , xo + ) na isqei |f(x) f(xo)| < . All

|g(x) g(xo)| = | |f(x)| |f(xo)| |() |f(x) f(xo)| <

(h (*) enai sunpeia thc trigwnikc anisthtac) prgma pou deqnei ti h genai suneqc sto xo. 2

7

Prtash 6.19 (Snjesh suneqn sunartsewn) 'Estw xo X R, stw f : X R mia sunrthsh kai stw g : Y R mia llh sunrthshme f(X) Y . An h f enai suneqc (sto xo) kai h g enai suneqc (stof(xo)), tte h g f : X R enai suneqc (sto xo).

Apdeixh 'Estw > 0. Ja brome > 0 ste gia kje x X me |xxo| < na isqei |g(f(x)) g(f(xo))| < .

H g enai suneqc sto yo = f(xo): ra uprqei > 0 ste an y Y kai|y yo| < na isqei |g(y) g(yo)| < . (1)

H f enai suneqc sto xo ra uprqei > 0 ste an x X kai |xxo| < na isqei |f(x) f(xo)| < .

All ta y = f(x) kai yo = f(xo) ankoun sto f(X) Y kai |y yo| =|f(x) f(xo)| < , ra ap thn (1) qoume |g(f(x)) g(f(xo))| < .

Deterh apdeixh: 'Estw (xn) tuqaa akolouja me xn X ste xn xo.Jtoume yn = f(xn). H sunqeia thc f sto xo deqnei ti h akolouja (yn)sugklnei sto yo = f(xo). All ap th sunqeia thc g sto yo qoume trag(yn) g(yo).

Epomnwc gia kje akolouja (xn) me xn X ste xn xo dexame ti(g f)(xn) = g(f(xn)) g(f(xo)). 2

8

7 Basik Jewrmata sunqeiac

Jerhma 7.1 An h f : [a, b] R enai suneqc tte enai nw fragmnhsto [a, b], dhl. uprqei M ste f(x) M gia kje x [a, b].

Apdeixh An h f den enai nw fragmnh sto [a, b], tte gia kje n N (oarijmc n den enai nw frgma tou sunlou f([a, b]) ra) uprqei xn [a, b]ste f(xn) > n. Dhmiourgetai tsi mia akolouja (xn) h opoa enai fragm-nh, giat a xn b gia kje n. Epomnwc (Jerhma Bolzano-Weierstrass)h (xn) qei mia upakolouja, stw (xkn), pou sugklnei se kpoio shmeo tou[a, b]. Ma tte, efson h f enai suneqc, h akolouja (f(xkn)) ja sugkl-nei. 'Omwc f(xkn) > kn gia kje n, ra h (f(xkn)) den enai kan fragmnh(jumzoume ti kn +). 2

H sunqeia thc f den mpore na paraleifje. Pardeigma:

f : [0, 1] R pou f(x) ={

1/x an x 6= 00 an x = 0

Epshc to sumprasma den isqei pnta, an to pedo orismo thc f den enaikleist kai fragmno disthma. Paradegmata:

f : (0, 1] R pou f(x) = 1x.

H f enai suneqc, orismnh se fragmno disthma, qi mwc kleist. Denenai nw fragmnh.

f : R+ R pou f(x) = x2.

H f enai suneqc, orismnh se disthma, qi mwc fragmno. Den enai nwfragmnh.

Jerhma 7.2 An f : [a, b] R enai suneqc tte qei mgisto sto [a, b],dhl. uprqei (toulqiston na) xo [a, b] ste f(xo) f(x) gia kje x [a, b].

Apdeixh Efson h f enai suneqc, ap to Jerhma 7.1 ja enai nwfragmnh, dhlad to f([a, b]) ja enai nw fragmno, ra ap thn plhrthta(!) tou R ja qei elqisto nw frgma, stw c. Uprqei, pwc qoumedexei, mia akolouja (sn) sto f([a, b]) ste sn sup f([a, b]) = c. Afo

1

sn f([a, b]), uprqei tn [a, b] ste f(tn) sup f([a, b]) = c. H akolouja(tn) enai fragmnh (a tn b gia kje n N) ra qei mia upakolouja1,stw (tkn), pou sugklnei se kpoio shmeo xo tou [a, b]. Efson h f enaisuneqc, h akolouja (f(tkn)) sugklnei sto f(xo). Ap thn llh meri epeidh akolouja (f(tn)) sugklnei sto c, kai h upakolouja thc (f(tkn)) prpei nasugklnei sto dio shmeo. 'Ara f(xo) = c kai sunepc f(xo) f(x) gia kjex [a, b].Deterh apdeixh An h f den qei mgisto sto [a, b], tte ja isqei f(x) 0, uprqei y [a, b)ste f(s) > 0 gia kje s [y, b]. To x1 dhlad ankei sto snolo

A = {x [a, b] : f |[a,x] < 0}

en to y den ankei sto snolo aut. To A enai mh ken kai nw fragmno(ap to y). Epomnwc, ap thn plhrthta tou R, qei elqisto nw frgma,

3

stw xo (a, b). Ja dexoume ti f(xo) = 0, apokleontac tic llec doperiptseic.

Prgmati, an upojsoume ti f(xo) < 0, tte ap thn sunqeia thc f stoxo ja uprqei mia perioq (xo , xo + ) [a, b] tou xo pou h f parneimnon arnhtikc timc. Efson xo = supA, uprqei x A me xo < x.H f loipn parnei arnhtikc mnon timc sto disthma [a, x] kajc kai stodisthma (xo , xo + 2 ], ra parnei arnhtikc mnon timc sthn nws touc,dhl. sto disthma [a, xo+

2]. Ma tte xo+

2 A, prgma pou antibanei sto

gegonc ti to xo enai nw frgma tou A. Apoklesjhke loipn h perptwshf(xo) < 0.

An pli upojsoume ti f(xo) > 0, tte ap thn sunqeia thc f sto xoja uprqei mia perioq (xo , xo + ) [a, b] tou xo pou h f parnei mnojetikc timc. All afo xo = supA, uprqei u A me xo < u. Ap thmi meri qoume u A, ra f(u) < 0, kai ap thn llh u (xo , xo + ),ra f(u) > 0. H antfash aut apokleei kai thn perptwsh f(xo) > 0. 2

H sunqeia thc f den mpore na paraleifje. Pardeigma: h sunrthsh

f : [0, 2] R pou f(x) ={1 an 0 x w > f(b),uprqei x [a, b] ste f(x) = w.

Apdeixh Efrmose to Jerhma 7.3 sthn g = w f . 2

Ac jumsoume ti na uposnolo I R lgetai disthma an periqei kjeshmeo pou enai metax do shmewn tou, dhlad an ikanopoie:

x, y I, x z y z I.

4

Jerhma 7.6 (Jerhma endiamswn timn) An I R enai di-sthma kai h f orzetai kai enai suneqc sto I, tte h eikna f(I) enai epshcdisthma.

Apdeixh Prpei na deiqje ti an c, d f(I) ma c 6= d, tte kje y metaxtwn c kai d ankei epshc sto f(I). Uprqoun a, b I ste f(a) = c kaif(b) = d. Upojtoume ti a < b (allic jewrome to disthma [b, a] anttou [a, b]). Tte h f enai orismnh kai suneqc sto [a, b]. An c = f(a) f(b) = d, to Prisma 7.5 odhge sto dio sumprasma. 2

Paratrhsh 7.7 To Jerhma den lei ti to f(I) enai kleist fragmnodisthma. (Pardeigma: an I = (0, 1] kai f : I R : x 1/x, ttef(I) = [1,+)). Epshc, an to I qei kra a, b, ta f(a), f(b) den enai pntakra tou f(I). (Pardeigma: an I = [0, 2] kai f : I R : x sin x, ttef(I) = [1, 1]).

Prisma 7.8 An h f : [a, b] R enai suneqc tte enai ktw fragmnhsto [a, b], dhl. uprqei ste f(x) gia kje x [a, b]. Mlista h flambnei elqisth tim sto [a, b], dhl. uprqei (toulqiston na) xo [a, b]ste f(xo) f(x) gia kje x [a, b].

Apdeixh Efrmose ta Jewrmata 7.1 kai 7.2 sthn f .

Prisma 7.9 An h f : [a, b] R enai suneqc tte enai fragmnh sto[a, b], dhl. uprqei K ste |f(x)| K gia kje x [a, b].

Prisma 7.10 An h f : [a, b] R enai suneqc tte to f([a, b]) enaikleist (kai fragmno) disthma. Mlista f([a, b]) = [m,M ], pou m =min{f(x) : x [a, b]} kai M = max{f(x) : x [a, b]}.

Apdeixh Ap to Jerhma 7.2 kai to Prisma 7.8, h f lambnei mgisthkai elqisth tim, m kai M , sto [a, b], opte m,M f([a, b]) kai f([a, b]) [m,M ]. Ap to Jerhma Endiamswn timn, to f([a, b]) enai disthma, kaiafo periqei ta m kai M ja enai anagkastik so me [m,M ]. 2

Efarmogc

5

Prtash 7.11 Kje jetikc arijmc a qei mia n-ost rza, uprqei dhladx R ste xn = a. Mlista an o n enai perittc, kje pragmatikc arijmca qei mia n-ost rza.

Apdeixh (i) 'Estw f : R+ R : f(x) = xn. 'Estw a > 0. Prpei nadeiqje ti uprqei x > 0 ste f(x) = a. Efson h f enai suneqc kai to R+enai disthma, ap to Jerhma 7.6 to f(R+) ja enai disthma, kai bebawcperiqei to 0. Gia na dexoume loipn ti periqei to a, arke na dexoume tiperiqei kpoio c > a. Uprqei mwc b > 0 ste f(b) > a (prgmati an a > 1pre p.q. b = a kai an a < 1 pre p.q. b = 1).

(ii) An o n enai perittc kai a < 0, ap to (i) uprqei x ste xn = a.All tte (x)n = xn = a.

Prtash 7.12 'Estw f : [0, 1] [0, 1] suneqc sunrthsh. Tte uprqeixo [0, 1] ste f(xo) = xo.

Apdeixh Jewrome th suneq sunrthsh g : [0, 1] R me g(x) = f(x)x.Efson 0 f(x) 1 gia kje x [0, 1], qoume g(0) 0 kai g(1) 0. Ang(0) = 0 g(1) = 0, teleisame. An qi, opte g(1) < 0 < g(0), ap toJerhma 7.3 uprqei xo (0, 1) ste g(xo) = 0. 2

Jerhma 7.13 'Estw I R na disthma kai f : I R suneqc kai 1-1.Tte

1. H sunrthsh f enai gnhswc montonh.

2. H antstrofh sunrthsh f1 enai suneqc.

H gnsia monotona thc f enai ekolh sunpeia twn do lhmmtwn pou ako-loujon:

Lmma 7.14 'Estw I R na disthma kai f : I R suneqc kai 1-1. Ana, b, c I me a f(b) > f(c).

Apdeixh Efson f(a) 6= f(b), mporome na upojsoume ti f(a) < f(b)(allic, jewrome thn f). Ja dexoume ti f(a) < f(b) < f(c). Ap toJerhma 7.6, to snolo f([a, b]) enai disthma kai afo periqei ta a kai b,ja periqei kai la ta endimesa shmea. An loipn f(a) < f(c) < f(b), tteja uprqei x [a, b] ste f(x) = f(c). Aut mwc apokleetai afo h f enai1-1, en x 6= c.

An pli f(c) < f(a) < f(b) tte ap to Jerhma 7.6 ja uprqei y [b, c]me f(y) = f(a), prgma pou epshc apokleetai.

6

Lmma 7.15 'Estw I R na disthma kai f : I R suneqc kai 1-1.An a, b, c, d I me a f(b) > f(c) > f(d).

Apdeixh Efrmose to prohgomeno Lmma stic tridec (a, b, c) kai (b, c, d).Ap thn prth trida, ja qoume f(a) < f(b) < f(c) f(a) > f(b) > f(c).An f(a) < f(b) < f(c), tte f(b) < f(c) ra ap thn deterh trida brskoumef(b) < f(c) < f(d). An pli f(a) > f(b) > f(c), tte f(b) > f(c) raf(b) > f(c) > f(d).

Apdeixh tou Jewrmatoc 7.13 (i) MonotonaStajeropoiome do shmea a < b sto I. Efson f(a) 6= f(b), mporome

na upojsoume ti f(a) < f(b) (allic, jewrome thn f).Prpei tra na dexoume ti h f enai gnhswc axousa, dhlad ti

x, y I, x < y f(x) < f(y).

An x = a kai y = b den qw tpote na apodexw. Allic, poia jsh ditaxhcki an qoun ta x, y sthn tetrda a, b, x, y, h dia jsh ja diathretai sticeiknec f(a), f(b), f(x), f(y) (ap to Lmma 7.15, to Lmma 7.14, an kpoioap ta x, y ankei sto {a, b}). Gia pardeigma an x < a < y < b ttef(x) < f(a) < f(y) < f(b) ra f(x) < f(y). An a < x = b < y ttef(a) < f(x) = f(b) < f(y) ra pli f(x) < f(y), kai otw kajexc.

(ii) Sunqeia thc f1.'Eqoume dexei tra ti h f enai gnhswc montonh. Upojtoume ti enai

axousa (allic jewrome thn f). Jtoume J f(I). 'Estw yo Jkai xo = f1(yo). An doje > 0 prpei na breje > 0 ste to snolo(yo , yo + ) J na apeikonzetai msa sto (xo , xo + ) ap thn f1.

Ac upojsoume prta ti to xo den enai kro tou I. Uprqei tte0 < ste (xo, xo+) I. An a = f(xo) kai b = f(xo+) ttea, b J kai a < yo 0 ste (yo , yo + ) (a, b) (p.q. = min{yo a, b yo}). Tte gia kje y (yo , yo + ) J qoume

a = f(xo ) < y < f(xo + ) = b

kai ra, efson h f1 enai gnhswc axousa3

xo < f1(y) < xo +

3an u < v tte f1(u) < f1(v), giat an f1(u) f1(v) ja eqame u v

7

dhlad |f1(y) f1(yo)| = |f1(y) xo| < .An pli to xo enai to ktw kro tou I (opte to yo enai to ktw kro

tou J), tte uprqei 0 < ste [xo, xo + ) I opte jtontac = f(xo +

) yo qoume gia kje y (yo , yo + ) J ti

yo y < yo + = f(xo + )

raxo < xo f1(y) < xo +

kai ra |f1(y) f1(yo)| = |f1(y) xo| < . 'Omoia antimetwpzetaikai h perptwsh pou to xo enai to nw kro tou I.

Paratrhsh 7.16 Parathrome ti sthn apdeixh thc sunqeiac thc f1

den qrhsimopoijhke h sunqeia thc f , all mnon h gnsia monotona thc kaito gegonc ti orzetai se disthma.

8

8 'Oria sunartsewn

Sthn pargrafo aut ja orsoume austhr t shmanei {to rio thc f(x) kajcto x tenei (plhsizei) sto xo}. Gia na qei nnoia to rio, prpei na mporomena plhsisoume osodpote kont sto xo, me shmea tou pedou orismo X thcf , diaforetik ap to xo. Prpei loipn prta na dieukrinsoume t shmanei{to xo enai shmeo sussreushc tou X}.

8.1 Shmea sussreushc sunlwn

Orismc 8.1 'Estw X R kai stw xo R. To xo lgetai shmeosussreushc tou X an kje perioq (xo , xo + ) tou xo periqei na(toulqiston) shmeo tou X ektc tou xo, dhlad an gia kje > 0 uprqeix X me 0 < |x xo| < .

Dhlad to xo enai shmeo sussreushc tou X an mpore na proseggisjeosodpote kont ap shmea tou X \ {xo}. To dio to xo mpore na ankei na mhn ankei sto X.

Orismc 8.2 An X R kai xo na shmeo tou X, tte to xo lgetai me-monwmno shmeo tou X an den enai shmeo sussreushc tou X, dhl.an uprqei > 0 ste (xo , xo + ) X = {xo}.

Pardeigma 8.1 An X = [a, b], kje shmeo tou X enai shmeo suss-reushc. An X = (a, b], kje shmeo tou X enai shmeo sussreushc, kaiepshc to a enai shmeo sussreushc tou X.

To snolo X = {1, 2, 3, . . .} den qei kanna shmeo sussreushc. Epo-mnwc la ta shmea tou X enai memonwmna.

An X = {1, 12, 1

3, . . .} tte la ta shmea tou X enai memonwmna, kai to

0 enai to mno shmeo sussreushc tou X.An X = {(1)n + 1

n: n = 1, 2, . . .} tte la ta shmea tou X enai

memonwmna, kai ta 1,1 enai ta mna shmea sussreushc tou X.

An to xo enai shmeo sussreushc touX, kje perioq tou periqei qi mnonna, all peira shmea tou X \ {xo}. Mlista, uprqei akolouja shmewntou X \ {xo} pou sugklnei sto xo:

Prtash 8.2 'Estw X R kai stw xo R. Ta aklouja enai isodnama:(i) To xo enai shmeo sussreushc tou X.

1

(ii) Kje perioq (xo , xo + ) tou xo periqei peira shmea tou X.(iii) Uprqei mia akolouja (xn) me xn X gia kje n kai xn 6= xm tan

n 6= m, ste xn xo.

Apdeixh (i) (ii) Kje perioq tou xo periqei shmea tou X diaforetikap to xo. An gia kpoio > 0 uprqe mno peperasmno pljoc shmewntou X sthn perioq (xo , xo + ), na (toulqiston) ap aut ja tanplhsistero sto xo. An 1 > 0 tan h apstash tou shmeou auto apto xo, tte sthn perioq (xo 1, xo + 1) den ja uprqe kanna shmeo touX \ {xo}, opte to xo den ja tan shmeo sussreushc tou X.

(ii) (iii) Kataskeuzoume epagwgik thn akolouja (xn) ap stoiqeatou X ste xn (xo 1n , xo +

1n) gia kje n (opte sgoura ja qoume xn

xo) kai frontzontac ste kje xn na enai diaforetik ap ta prohgomena:Afo sthn perioq (xo 1, xo + 1) uprqoun peira stoiqea tou X, ja

uprqei na, stw x1, diaforetik ap'to xo.Afo sthn perioq (xo 12 , xo +

12) uprqoun peira stoiqea tou X, ja

uprqei na, stw x2, diaforetik ap'to xo kai ap to x1.An qoume kataskeusei x1, x2, . . . , xn1 X diaforetik metax touc

(kai ap to xo) ste xk (xo 1k , xo +1k) gia k = 1, 2, . . . , n 1, afo sthn

perioq (xo 1n , xo +1n) uprqoun peira stoiqea tou X, ja uprqei na,

stw xn, diaforetik ap' la ta prohgomena.'Etsi oloklhrjhke h epagwgik kataskeu thc akoloujac (xn).

(iii) (i) An uprqei akolouja (xn) diaforetikn an do shmewn touX pou sugklnei sto xo, tte gia kje > 0 ja uprqei no N ste|xn xo| < gia kje n no, kai afo oi roi thc akoloujac enai diafore-tiko, kpoioc ap autoc ja enai diaforetikc ap'to xo, opte ja ankei stosnolo (xo , xo + ) X \ {xo}. 2

Prtash 8.3 'Estw X R kai f : X R mia sunrthsh. An xo Xenai memonwmno shmeo, tte h f enai suneqc sto xo.

Apdeixh 'Estw > 0. Ja br > 0 ste gia kje x X me |x xo| < na isqei |f(x) f(xo)| < .

Ma afo to xo enai memonwmno shmeo tou X, uprqei > 0 ste tomno x X pou ikanopoie |x xo| < na enai to dio to xo. Epomnwc anx X kai |x xo| < tte x = xo opte |f(x) f(xo)| = 0 < . 2

2

Prtash 8.4 'Estw X R kai f, g : X R do suneqec sunartseic.An xo X enai shmeo sussreushc tou X kai f(x) = g(x) gia kje x X\{xo}, tte f(xo) = g(xo).

Apdeixh 'Estw (xn) akolouja stoiqewn tou X \ {xo} pou sugklnei stoxo (afo to xo enai shmeo sussreushc tou X, uprqei ttoia akolouja).Afo oi f kai g enai suneqec, qoume f(xn) f(xo) kai g(xn) g(xo).All xn X \ {xo}, ra f(xn) = g(xn) gia kje n, kai sunepc

f(xo) = limn

f(xn) = limn

g(xn) = g(xo). 2

8.2 'Oria sunartsewn

Orismc 8.3 'Estw X R, stw f : X R mia sunrthsh kai xo Rshmeo sussreushc tou X (to xo mpore na ankei na mhn ankei sto pedoorismo X thc f).

(i) Lme ti to rio thc f kajc to x xo uprqei kai isotaime ton arijm yo R an:

Gia kje > 0 uprqei > 0 stean x X kai 0 < |x xo| < tte |f(x) yo| < .

Grfoume tte limxxo

f(x) = yo.

(ii) Lme ti h f tenei sto + kajc to x xo an:Gia kje M > 0 uprqei > 0 ste

an x X kai 0 < |x xo| < tte f(x) > M .(iii) Lme ti h f tenei sto kajc to x xo an:

Gia kje M < 0 uprqei > 0 stean x X kai 0 < |x xo| < tte f(x) < M .

Shmeinoume ti h parxh tou orou qei sqsh me thn sumperifor thc fkont sto xo. Epomnwc, propjesh gia thn parxh tou orou enai to xona enai shmeo sussreushc tou pedou orismo X thc sunrthshc. Sto dioto xo h f mpore na mhn orzetai kajlou. All kai an orzetai, h tim f(xo)den endiafrei son afor sthn parxh tou orou.

Paradegmata 8.5 (i) limx3 x2 = 9.

(ii) An f : R R me f(x) ={

x2 an x 6= 30 an x = 3

, to rio limx3 f(x)

uprqei kai isotai me 9.

3

(iii) To rio limx0 1x den uprqei.(iv) An f : (0,+) R me f(x) = 1

x, tte limx0 f(x) = +.

(v) To rio limx0 sin 1x den uprqei.

(vi) An f : [0, 1] R me f(x) ={

0 an x rrhtoc1n

an x = mnme (m,n) = 1

tte gia kje xo [0, 1] to rio limxxo f(x) uprqei kai isotai me 0.

Apdeixh Ta (i), (ii) kai (iv) apotelon aplc efarmogc tou orismo. Ta(iii) kai (v) apodeiknontai eukoltera me qrsh thc arqc thc metaforc(bl. paraktw).Apdeixh tou (vi) 'Estw > 0. Ja dexw ti, gia kje x [0, 1] {arketkont} sto xo, all me x 6= xo, isqei ti |f(x) 0| < . Ac dialxoume nanfusik arijm k ste 1

k . An gia kpoio x [0, 1] den isqei h sqsh

|f(x) 0| < , tte isqei f(x) 1k, ra (ap ton orism thc f) o x ja enai

angwgo klsma mnme paronomast mikrtero so me k. Dhlad oi timc

tou x pou prpei na apoklesoume enai

1,1

2,1

3,2

3,1

4,3

4, . . . ,

1

k, . . . ,

k 1k

.

To pljoc autn twn arijmn enai peperasmno, epomnwc uprqei mia perio-q (xo , xo + ) pou den periqei kannan apautoc, ektc (endeqomnwc)ap to xo. 'Ara gia kje x [0, 1] me 0 < |xxo| < ja qoume anagkastik|f(x)| < 1

k, ra |f(x) 0| < . 2

Prtash 8.6 (Arq thc metaforc) 'Estw X R, stw f : X Rmia sunrthsh, xo R na shmeo sussreushc tou X kai yo R. Tte tolimxxo

f(x) uprqei kai isotai me yo an kai mnon an gia kje akolouja

(xn) me xn X\{xo} gia kje n ttoia ste limn

xn = xo, h akolouja

(f(xn)) sugklnei, kai mlista sto yo.

Apdeixh Upojtoume prta ti to limxxo

f(x) uprqei kai isotai me yo.

Tte gia kje > 0 ja uprqei > 0 ste an x X kai 0 < |x xo| < naisqei |f(x) yo| < .

'Estw (xn) mia (tuqaa) akolouja me xn X\{xo} gia kje n, pou sug-klnei sto xo. Ja uprqei tte no N ste |xn xo| < gia kje n no.Mlista efson xn 6= xo ja qoume 0 < |xn xo| < . Epomnwc ap thn

4

prohgomenh pargrafo qoume |f(xn) yo| < gia kje n no. Afo to enai aujareto, dexame ti h akolouja (f(xn)) sugklnei sto yo.

'Estw antstrofa ti, gia kje akolouja (xn) me xn X\{xo} pousugklnei sto xo, h akolouja (f(xn)) sugklnei sto yo. Ja dexoume tte tito lim

xxof(x) uprqei kai isotai me yo.

Prgmati an aut den isqei tte ja uprqei kpoioc jetikc arijmc ste gia kanna > 0 na mhn isqei h sunepagwg

{an x X kai 0 < |x xo| < tte |f(x) yo| < }.Tte gia kje thc morfc = 1

n

ja uprqei xn X me 0 < |xn xo| < 1n all |f(xn) yo| .Sqhmatzoume thn akolouja (xn): 'Eqoume xn X kai xn 6= xo (efson

0 < |xn xo|) gia kje n kai xn xo (efson |xn xo| < 1n gia kje n).All |f(xn)yo| gia kje n, epomnwc h akolouja (f(xn)) den sugklneisto yo. 2

Paratrhsh Den arke gia thn parxh tou orou limxxo

f(x) na isqei h

sunjkh thc prtashc gia kpoia akolouja (xn). Prpei na isqei gia lectic akoloujec pou ikanopoion xn X\{xo} gia kje n kai lim

nxn = xo.

Gia pardeigma an f enai h sunrthsh Dirichlet (f(q) = 1 an q rhtc,

f(t) = 0 an t rrhtoc) tte limn

f(1

n) = 1, all to rio lim

x0f(x) den uprqei

(giat?). To dio sumbanei kai sthn epmenh efarmog.

Efarmog Ta ria limx0

1

xkai lim

x0sin

1

xden uprqoun.

Apdeixh Jewrome tic akoloujec xn = 1n kai yn =1

22n (n = 1, 2, . . .).

Parathrome ti xn 6= 0 6= yn gia kje n kai ti limn xn = 0 = limn yn.'Omwc an f(x) = 1

xkai g(x) = sin 1

x(x 6= 0), tte ta ria limn f(xn) kai

limn f(yn) enai diaforetik metax touc (enai + kai antstoiqa). E-pshc ta ria limn g(xn) kai limn g(yn) enai diaforetik metax touc (enai 0kai 1 antstoiqa).

Prtash 8.7 'Estw X R, stw f : X R mia sunrthsh kai stwxo X na shmeo sussreushc tou X. Tte h f enai suneqc sto xo ankai mnon an to lim

xxof(x) uprqei kai isotai me f(xo).

Apdeixh An h f enai suneqc sto xo tte gia kje > 0 ja uprqei > 0ste an x X kai |x xo| < na isqei |f(x) f(xo)| < . Epomnwc, an

5

x X kai 0 < |x xo| < tte |f(x) f(xo)| < , ra to limxxo

f(x) uprqei

kai isotai me f(xo).An antstrofa to lim

xxof(x) uprqei kai isotai me f(xo), tte gia kje

> 0 ja uprqei > 0 ste an x X kai 0 < |x xo| < na isqei |f(x)f(xo)| < . Epomnwc an x X kai |x xo| < ja isqei |f(x) f(xo)| < (giat an |x xo| = 0 tte |f(x) f(xo)| = 0 < ). 'Ara h f ja enai suneqcsto xo. 2

Shmewsh Tonzoume kai pli ti h Prtash qei efarmog mnon tanto xo enai shmeo sussreushc tou pedou orismo thc sunrthshc. Se k-je memonwmno shmeo so tou pedou orismo thc h sunrthsh enai suneqc(Prtash 8.3), all to rio lim

xsof(x) den qei nnoia.

Paratrhsh 8.8 An efarmsoume thn Prtash 8.7 sto Pardeigma 8.5(vi) prokptei ti se kje rrhto xo [0, 1] h sunrthsh f enai suneqc(afo to lim

xxof(x) uprqei kai isotai me 0 = f(xo)) en enai asuneqc se

kje rht mn (0, 1] (afo to lim

xm/nf(x) uprqei men, all den isotai me

f(mn)).

H epmenh Paratrhsh bebainei ti to rio limxxo

f(x) uprqei an kai

mnon an h f mpore na orisje sto xo tsi ste na enai suneqc sto xo.H Paratrhsh aut mpore na qrhsimesei wc isodnamoc orismc tou oroulimxxo

f(x).

Paratrhsh 8.9 'Estw X R, stw f : X R mia sunrthsh kai stwxo R na shmeo sussreushc tou X. Tte to lim

xxof(x) uprqei kai isotai

me yo an kai mnon an h sunrthsh

f : X {xo} R me f(x) ={

f(x), an x X\{xo}yo, an x = xo

enai suneqc sto xo.

Apdeixh Orzoume th sunrthsh f pwc sthn ekfnhsh. Smfwna methn Prtash 8.7, h f enai suneqc sto xo an kai mnon an to rio lim

xxof(x)

uprqei kai isotai me f(xo) = yo. Efson mwc f(x) = f(x) gia kje x X\{xo}, to lim

xxof(x) uprqei kai isotai me yo an kai mnon an to lim

xxof(x)

uprqei kai isotai me yo. 2

6

Prtash 8.10 'Estw X R, stw xo R na shmeo sussreushc touX, stw f, g : X R sunartseic kai R. An ta ria lim

xxof(x) = yo kai

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APEIROSTIKOS LOGISMOS ApI_katav.pdf