Post on 02-Oct-2021
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