Lecture G

KatyCraig 2021

Recall

Limit Theorems

Timllimit of sum is sum of limits If snandtn are

convergentsequences n'Ia lsnttnl

ninfosnthifatn.FMlimit ofproduct is product of limits If Sm andtn are convergent sequences nFa sntnfhifasnkhinfa.tn

Them limit ofquotient is quotientof limits If snand tn

are convergent sequences Snf 0 for all nand n Fa Sn 0 then

hiFa stun hirsuteI imn N Sn

Basi cExamplesThin basic examplesa tinF th P 0 if p O

b info an 0 if lak Ic info n'In 1d Is a'In if a O

Divergence to to

Defklivergestotoor o A sequence sn divergesto to if for all m 0 there exists N

so that n N ensures Sns M We write n Fosner

Likewise a sequence Sn diverges to if forall m 0 there exists N so that n Nensures Sn LM

The Suppose linFosse to and linFotn 0

Then times Sntn t N

Tim suppose son is a sequence of positivereal numbers Then info sn ta linFasteo

Three If Linfu Sn ta then limo Sn o

So far we have studied convergent divergentand bounded sequences There are twomore important types ofsequences monotoneand Cauchy sequences

Def increasingKlecreasinglmonotone sequencesA sequence Sn is

increasingif Snesnt Vn L

A sequence Sn isdecreasing

if Snzsnt Hn

A sequence Sn is monotone if it is either nyotmon

increasing or decreasing IIet Hint's LEINE III I Ethel Edith

Ex an I IEM is increasingat

birth isincreasingcn C 1

nis not monotonic

If mzn then m n t K for KEN K2 0 It suffices to showSnESmtK for all KE Z K 0 Basecase k o is true since she SntoInducetivestep Assume SnESntk Bydefnofincreasing Sntk cShtetl So sheSuttaRemark If Sn is increasing thenSn E Sm whenever n C m

MAJOR THEOREM 3

Thin All bounded monotone sequences converge

Mental imageMinn 7pls r

i 5 swine IN

ii g n

Pf Suppose Sn is a bounded increasingsequence Define 5 Srine INSince Su is a bounded emoji'eFiEeYn havethat S is a bottledFEE Bydefnof RS has a supremum We will showlines sn sup S

Fix E 0 Since sup s is an upper boundfor S supG Z Sn f me IN and

SopGt E Sn Fn EIN Ct

Since supls is the least upper boundsupls E is not an upper bound of Sthat is there exists NEIN s't

SN supls E Since Su is increasingSn sup CS E f n N Cta

Combining Hdt we have An N

Isn sup s IL E Since E 0 was arbitraryinfo Sn sup s

Now suppose Sn is a bounded decreasingsequence Then Sn is a bounded incr

sequence Thus info since IR By limittheorem L'Fo Sn IES n l D c HenceSn converges D

yesIn fact even unbounded monotone

sequences always have a limit

Them If Sn is an unbounded increasingsequence then I im sn to

If Sn is an

unboundeddecreasing sequence then limsse

x

Pf Suppose Sn is an unbounded increasingsnksequence Fix M 0 Since Su is unboundedand Su is increasing 5 sn i n e INis not bounded above Thus F N s t

SN M Since Sin is increasing Sn SN MV n N Since M 0 was arbitrary thisshows Linfs Sn to

Now suppose sn is an unbounded decreasingsequence Then Sn is an unbounded

increasing sequence so n Fa Sn to

thus by theorem from last class n Fo sie a

Sn is unbold a bore ifS Sn ne IN doesn'thaIn

summary if Sn is monotone an upperbound

Jthing sniff

r SE R i f Sn isbdd_if Sn is unbddabove

so if Sn is unbdd below

therefore for any monotone sequence snim Sn exists

In general we can think of plenty examplesof sequences whose limits don't exist

It turns out that there is a generalizationof the idea of limit that does always exist

This should remindyou of

the fact thatthe maximum of a set 5 doesn'talwaysexist but sopCS always exist

Defclimsoplliming For any sequencesn

RUEtN3

line B Sn loins sup Sm n N

limniff sn In infssi.noBbNElRUEo3SnoeoyA1of

a 2 93 94 a9 a o

bbibs

5

oo of a a

0 OO o

0 OFo t o bg bz bg

I l l I i f l I I n12 3 4 5 6 10