MATH 2050C Lecture 15 Mar 9

Reminder Midterm this Thursday Friday

Last time Cauchysequence

Def kn Cauchy iff VE so 7 H H E CIN st ldn XmIc E th m 1

Remark One doNOT need to refer to a limit awe k in this deft

1hm Cauchy criteria Ku Cauchy 7 can convergent

Example Let xn be the sequence defined by

X a 1 Xz 2 Xu a Iz Ka i Xu 2 V n 33

Show that Gcn is convergent and find him Xu

Think kn 1 2 1.5 1.75 1 625

bold Not monotone

Pf By M.I Exercise we have

1 E Xu E 2 th C IN

Kuei Xu I2

V n C IN

claim xn is Cauchy

Pf of Claim Let E so be fixed but arbitrary4

Choose H EIN St I I E

Then V m n 3 It we want to show

1 Xm kn l c E t m n H

W L O G assume M s n z H

Xm Xn I E I Xm Xm i l t l2cm i Xm al t lKuei Xu I

t t 1

l It t t2

2 E 4 IH E 4 s E

0

ByCauchy Criteria lim kn X exists

Consider the subof 2k KEN tTake n soo

Ncte olim Kaci XKia X 2C t K X

pXue I Iz t f t Jst t 5 Lyfe

It I

l L4

Take K soo we have X It 1 53

is

Course Outline

real number system IR

µ limit of seq xnPreparations

limit of function ie him fix L17 xia

continuity of functions

Limit of Functions Ch 4 in textbook

LGOAL Define him fix for functions f A EIR IR

x2C

him f X for those c s which arewe shall only define x c

cluster point of A so fCx isdefinedE s

IDEA fix L when XTC and X C A

Def't i Lef A E IR We say that C E IR is a cluster pointof A

iff f 8 so I X E A St Xtc and Ix C Is S

Ac C

exo ox 2 Rp t

xx

to ARemark A cluster pt C E B may or may not belong

Examples

A f 1 2 N cluster pt K K cooks its1 2

A o 1 Any C C 0,1 is a cluster pt cc E ofcf sir1 2

A Ai An NI cluster ptA IN NI Chesterpt o o SIR

1 2 4 5 G n

A ht NEIN ONLY 1 ClusterptC

G O C do ifI z I

Prop C C IR is a cluster point of A

I seq au in A st An F C f n c IN

and him canI C

stuff Take S In bydefa 7 An C A sen

asn in

An F C and I an c l c Sn In o

b

Common mistake in Ex 33.7

2C a o

o

unlessK 0

We now state the most important definition for this chapter

Def'd Let f A EIR B be a function

Suppose C C IR is a cluster point of A

We say that f converges to L c IR at c written

a

him fCx L or fc L as x cx c

iff f E so I 8 8 E O s tso Xtc

floc L Is E V X E A where oe Ix c Is S

Example 1 Let f IR IR be FCK x for all KEIR

lim Tex c t c e CRx c

Pf Any C CSR is a cluster pt of A IR

Let E o be fixed but arbitrary

Choose 8 so se S E

THEN V K E 1B and 0 s I k C l C S we have

fca e l I x C l c S Eis

Rlm k Emc f x may exists with f beingdefined at C

EI f A Co 1 IR Fox x

lim fax I Ax 1

Example 2 him x 2 CZ i.ef A D A

fax x2x c

oooha

Iii iii e war

iiiiiiEIII.IENote Suppose I k c l C 1 then I 124114 Ix ctE 21418 S c e

Kl E X C1 t lol C 1 t ICIIX Cla s 1 1 a lot 18

Choose S min f 1E3

2 I121cL

THEN V KE A IR with 0 C Ix c l c 8 we have

1 2E El 1kt ol Ix c 1 E l 1214 S C E

Example 3 him to E where Ct ox c

Considering f A IR Io IR STK

NotesAny CG IR is a clusterPt of A oOg

Pf Let's assume C S OIf occx etc 8 then

i iifIx Cla Z SEE

124 7 0Need 1 1 bold awayfrom 0

Take 8 meh f E so

Then V X E A and o six c l c gf I T.la iR

we have

1 E I Ix ol Z S E E

is