Math Real Analysis

MATH 3001 W21 A1 Due o

n Fe brua

ry 3 rd

(1) Let {xn } be a real sequence . Show the following :

{Xn) is unbounded ⇐ there is a subsequence {Xn , }

at . lim Ku ,K-so I =D

(2) Find tho real sequences Exist, {un ) such that xn →x, yn →y and xn s yn hit x>y .

(3) Find the tail of the gents II. am, where a E C-bi ) . That d, calculate exphvitty a

En = I at

.

f-htt

(4) Suppose a sequence { xn) has two hitsquinces lack} and {bet such that every xn is either an 9k or a fee . Suppose that iahjmak = a and effy be =p . show that the sets

of limit pouts of Hn) equals S= {a, pig .

(5) Let Am = FIX " , XE tht ) . Prove that {Amy is a Cauchy sequence .

MATH 3001 W21 S1

H ⇐) Ik sit . HH >I (because {Xn } is unbounded)I 2 Then among all indices k>he, Eh, at

. 14,132 ( if that was wet the case

, then fling would be founded !) . Then

annoy all k>kz I kg sit . Hk, 133 . ETC. So : HAEIN

E kn Sot . lxknl > '

n . Also , Keck! kzs

- - '

. Thus fkn ,4,

is a subsequence of fxnlgn with him Hn,d=N K

⇐) TM>o E R sit . Kyl>M (as Ha! -9%) . Hence {xn4 cannot be bounded .

(2) Xn= - th , yn = th

, X=y=o .

③ Am = FI,at = It at . - +am = I - am "

i -q

-

A- = ¥n,pAm= Fa = II. ai

En = A - An

=

9h41 I -

q '

④ By definition of the sets, we have {a, pig c S . Now we show that { a, pips

, namely that any LES must be equal to a or p .

Let xn , be a convergent subsequence of xn , with fruit L .

Each of the xn , is some as or some be Ifor some index s,t).

• suppose infinitely many of the xn, (as K varies) are from both

He bit of the as and the bee . Then both the 9k and be have to converge to L,

so a=p=L . ° Suppose the xn, contains only Amlely many of the ap . Hence

it

contains infinitely many of the be . So L=p o If Xnk conform Amlely many of the be only , then a=L .

(5) I Am-An 1=17×1 Iam"-x ""

I take m>n WoLOG

= III / km -n - it

E 1¥ , lxlh " "→

°

yo .