Question

4. (20 pts) Let {xn} be a Cauchy sequence. Show that a) (5 pts) {xn} is bounded. Hint: See Lecture 4 notes b) (5 pts) {Jxn} is a Cauchy sequence. Hint: Use the following inequality ||x| - |y|| < |x - y|, for all x, y E R. _ subsequence of {xn} and xn c) (5 pts) If {xnk} is a See Lecture 4 notes. as k - oo, then xn OO as n»oo. Hint: > d) (5 pts) If {yn} is a Cauchy sequence then {xn + yn} is a Cauchy sequence. 5a) (5 pts) Find lim inf (xn) and lim sup (xn), for xn = 4 + (-1)" (1 - ). Justify your answer = 3 and lim inf (x) 5b) (5 pts) Find a sequence xn with lim sup (rn =-2. 5c) (10 pts) Let {x,n} be a bounded sequence of real numbers with lim inf (xn) = x and lim sup (xn) = y, where , yER. Show that {xn} has subsequences {an} and {bn}, such that an -» x and bn y.

Answer 1

IF YOU HAVE ANY DOUBTS COMMENT BELOW I WILL BE TTHERE TO HELP YOU..ALL THE BEST..

AS tor given date sinac n a Cauhy geten so, fer grven E>o,3 a tve Xn-m indager Seh Hest Cauhy sevuen te and N n-m cohonever n,mzN be t endex -tot ten { NX+N / -X NXINXX (N X Mma EXN, 721 -- 12 Per alf m tnce o boun dec knoc Hat we X-y/for al R l1201m £ (Xn-Xm } )xn -120 E n m Cauchy eaence ノ スのkしa and メnと → af k→ So t aia that Cachy enone fe Tuh3 Xn d Gf n NOw m Suuh thou 9iven Eso ノ子atve 2 uRu 37I u- uh Scanned with CS CamScanner

for m m NOCD bnyn-Cum m) Cxn-m)tym#m) . uplwxUx 1 Cnn )- Cmm) <E nm TH uRU A H Exn tyn3auby ceucdry -Hen Ce Srene thank you Cs Scanned with CamScanner

AS FOR CHEG RULES WE ANSWER ONE QUESTION ONLY...

I HOPE YOU UNDERSTAND..

PLS RATE THUMBS UP..ITS HELPS ME ALOT..

THANK YOU...!!