2.13.4 2.13.5 Show that lim supno (-X) = -(liminf ,-Xn). If two sequences {an) and {bn}...
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...
please i need the question (m) (n)(o) for the detailed proof and example ! thanks ! Prove that the given statements are true concerning the two sequences of real numbers (an) and (b. 0 and limn→” an-L > 0, then (m) If an, bn lim an) (lim supbn) lim sup (anbn) - (n) If an > 0 and bn > 0 and if both lim supn→ooan and lim supn→oobn are either finite or infinite, then lim sup,-,(anbn) < (lim sup,-o...
18. If ai, az, as,... is a bounded sequence of real numbers, define lim sup an (also denoted lim an) to be --+ n+ l.u.b. {z ER: an > & for an infinite number of integers n} and define lim inf an (also denoted lim an) to be g.l.b. {ER: An <for an infinite number of integers n}. Prove that lim inf an Slim sup an, with the equality holding if and only if the sequence converges. 19. Let ai,...
ANSWER 1 & 2 please. Show work for my understanding and upvote. THANK YOU!! Problem 1. Let {x,n} and {yn} be two sequences of real numbers such that xn < Yn for all n E N are both convergent, then lim,,-t00 Xn < lim2+0 Yn (a) (2 pts) Prove that if {xn} and {yn} Hint: Apply the conclusion of Prob 3 (a) from HW3 on the sequence {yn - X'n}. are not necessarily convergent we still have: n+0 Yn and...
2. Let {An}n>1 and {Bn}n>ı be two sequences of measurable sets in the measurable space (12,F). Set Cn = An ñ Bn, Dn = An U Bn: (1) Show that (Tim An) ^ ( lim Bm) – lim Cn (lim An) ( lim Bu) C lim Dm and 100 noo (2) Show by example the two inclusions in (1) can be strict.
Question 16 (1 point) For two bounded sequences of real numbers {Xn } n=1 and {yn}"=1, if Xn > Yn for every n E N, then we have limsup Xn > liminf yn n+00 no O True O False
#s 2, 3, 6 2. Let (En)acy be a sequence in R (a) Show that xn → oo if and only if-An →-oo. (b) If xn > 0 for all n in N, show that linnAn = 0 if and only if lim-= oo. 3. Let ()nEN be a sequence in R. (a) If x <0 for all n in N, show that - -oo if and only if xl 0o. (b) Show, by example, that if kal → oo,...
5. Let {xn} and {yn} be sequences of real numbers such that x1 = 2 and y1 = 8 and for n = 1,2,3,··· x2nyn + xnyn2 x2n + yn2 xn+1 = x2 + y2 and yn+1 = x + y . nn nn (a) Prove that xn+1 − yn+1 = −(x3n − yn3 )(xn − yn) for all positive integers n. (xn +yn)(x2n +yn2) (b) Show that 0 < xn ≤ yn for all positive integers n. Hence, prove...
Exercise 2.3.7: Let {xn} and {yn} be bounded sequences. a) Show that {Xn+yn} is bounded. b) Show that (lim inf xn) + (lim inf yn) < lim inf (Xn tyn). noo Hint: Find a subsequence {Xn; +yn;} of {Xn +yn} that converges. Then find a subsequence {Xnm;} of {Xn;} that converges. Then apply what you know about limits. n->00 c) Find an explicit {{n} and {yn} such that noo (lim inf xn) + (lim inf yn) <lim inf (Xn+yn). noo...
Example: Let {xn} be a sequence of real numbers. Show that Proposition 0.1 1. If r is bounded above, x = lim sup (r) if and only if For all 0 there is an NEN, such that x <x+e whenevern > N, and b. For all >0 and all M, there is n > M with x - e< In a. Example: Let {xn} be a sequence of real numbers. Show that Proposition 0.1 1. If r is bounded above,...