We are given that (an)nen is a sequence of real numbers that is bounded from above....

Question

Question

We are given that (an)nen is a sequence of real numbers that is bounded from above. Now consider L:= lim supnoo An. Prove tha

Answer 1

Answer #1

an>2- ɛo fer only finite Melny liven (an) nein be any Reul sequeme which is bounded A som above suplan) exist. Now let Lo li

Answer 2

Similar Homework Help Questions

We are given that (an)nen is a sequence of real numbers that is bounded from above....

We are given that (an)nen is a sequence of real numbers that is bounded from above. Now consider L:= lim supnoo An. Prove that: For all € > 0 there are infinitely many n for which an > L - € AND For all e > 0 there are only finitely many n for which an > L te
1,2 Let (an)nen be a sequence of real numbers that is bounded from above. Consider L...

1,2 Let (an)nen be a sequence of real numbers that is bounded from above. Consider L := lim suPn7o An, prove that: For all e > 0 there are only finitely many n for which an > L + €. For all e > 0 there are infinitely many n for which an > L - €.
Example: Let {xn} be a sequence of real numbers. Show that Proposition 0.1 1. If r...

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,...

Let (an)nen be a bounded sequence in R. For all n e N define bn =...

Let (an)nen be a bounded sequence in R. For all n e N define bn = sup{am, On+1, On+2,...}. (You do not have to show that the supremum exists.) (a) Prove that the sequence (bn)nen is a monotone sequence. (b) Prove that the sequence (bn)nen is convergent. (c) Prove or disprove: lim an = lim bre. 100 000
18. If ai, az, as,... is a bounded sequence of real numbers, define lim sup an...

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,...
Let (xn) be a bounded sequence of real numbers, and put u = lim supn→∞ xn . Let E be the set consisting of the limits of all convergent subsequences of (xn). Show that u ∈ E and that u = sup(E). Form...

Let (xn) be a bounded sequence of real numbers, and put u = lim supn→∞ xn . Let E be the set consisting of the limits of all convergent subsequences of (xn). Show that u ∈ E and that u = sup(E). Formulate and prove a similar result for lim infn→∞ xn . Thank you! 7. Let (Fm) be a bounded sequence of real numbers, and put u-lim supn→oorn . Let E be the set consisting of the limits of...

1. Let {n} be a sequence of non negative real numbers, and suppose that limnan =...

1. Let {n} be a sequence of non negative real numbers, and suppose that limnan = 0 and 11 + x2 + ... + In <oo. lim sup - n-00 Prove that the sequence x + x + ... + converges and determine its limit. Hint: Start by trying to determine lim supno Yn. What can you say about lim infn- Yn? 3 ) for all n Expanded Hint: First, show that given any e > 0 we have (...
1. Let Xn ER be a sequence of real numbers. (a) Prove that if Xn is...

1. Let Xn ER be a sequence of real numbers. (a) Prove that if Xn is an increasing sequence bounded above, that is, if for all n, xn < Xn+1 and there exists M E R such that for all n E N, Xn < M, then limny Xn = sup{Xnin EN}. (b) Prove that if Xn is a decreasing sequence bounded below, that is, if for all n, Xn+1 < xn and there exists M ER such that for...
(1) Let a (.. ,a-2, a-1,ao, a1, a2,...) be a sequence of real numbers so that f(n) an. (We may eq...

(1) Let a (.. ,a-2, a-1,ao, a1, a2,...) be a sequence of real numbers so that f(n) an. (We may equivalently write a = (abez) Consider the homogeneous linear recurrence p(A)/(n) = (A2-A-1)/(n) = 0. (a) Show ak-2-ak-ak-1 for all k z. (b) When we let ao 0 and a 1 we arrive at our usual Fibonacci numbers, f However, given the result from (a) we many consider f-k where k0. Using the Principle of Strong Mathematical Induction slow j-,-(-1...

***You must follow the comments*** Topic: Mathematical Real Analysis - Let (xn) be a bounded sequence...

***You must follow the comments*** Topic: Mathematical Real Analysis - Let (xn) be a bounded sequence ((xn) is not necessarily convergent), and assume that yn → 0. Show that lim n→∞ (xnyn) = 0. Question1. All the solution state that there exists M >0 and xn<=M . My question is that why M always be bigger than 0 and Why it is bounded above ? why it is not m<=xn bounded below???? Question. 2. if the sequence is convergent, then...