Let (an) be a bounded sequence. (a) Prove that the sequence defined by yn = sup{ak k ≥n} converges. (b) The limit superior of (an), or lim sup an, is defined by lim sup an = lim yn where yn is the sequence from part (a) of this exercise. Provide a reason- able definition for lim inf an and briefly explain why it always exists for any bounded sequence. (c) Prove that lim inf an ≤ lim sup an for every bounded sequence, and give an example of a sequence for which the inequality is strict.

Q: Let (n) be a bounded sequence. (a) Prove that lim sup(x) = limm- (b) Prove that lim inf(x) = limm-…

A:

Q: Give an example of a sequence an whose set of subsequential limits S = {a €RU{-0}U{x} : a = lim ak,…

A:

Q: (a) Let (an) be a bounded (not necessarily convergent)sequence, and assume lim bn = 0. Show that…

A:

Q: Let x, = 3. Using the algebra of limits and standard results on limits to establish/justify if the…

A: Consider a sequence an. Then if the limit limn→∞an exist then the sequence converges otherwise the…

Q: (ii) Prove that if a sequence Xn converges to a limit I, then any subsequence of Xnalso converges to…

A:

Q: Prove that limx → c f(x) = ∞ if and only if for every sequence an that is in the domain of f and…

A: As per the question we have to prove that limx → c f(x) = ∞ if and only if for every sequence an…

Q: 6]. sequence to show that, lim Use the definition of limit of a (2n²-3 7n²+8 2/7

A: Definition of limit if a sequence  limit of a sequence is the value the sequence approaches as the…

Q: a) Suppose (pn) has limit L, and let S be the set of values of the terms of (pn). Can we say that L…

A:

Q: Let (an) be an unbounded sequence. Show that it has a subsequence (bk) = (ank) 1 such that lim = =…

A: Given, an  unbounded sequence.It is required to prove there exists a subsequence  such that .To…

Q: Use the definition of the limit of a sequence to prove that (-1)"n' lim = 0. n2 + 1

A: To prove limn→0(-1)nnn2+1=0

Q: Exercise 2.32. Find an example of a topological space and a convergent sequence in that space for…

A:

Q: If a sequence C₁, C2, C3,... has limit K then the sequence e limit of the sequence given by b₁ = (n)…

A:

Q: Find the limit of the sequence {√5. √9+√5. √3+ √5+ √5, √/3+ √/3— √9+√5, √3+√3+√3+ √9+√5. √√3+√√3,…

A:

Q: Let (an). (bn) be two bounded sequences. a. Show that lim sup max(a,, b,) > max(lim sup an, lim sup…

A: Given: Let's say there are two bounded sequences (an, bn) .a. Demonstrate that limn→∞sup max(an,…

Q: Exercise 4 We say that the sequence {a} is unbounded from above if there exists a subsequence (an,}…

A:

Q: ii) Prove that if a sequence XnXn converges to a limit l, then any subsequence of XnXnalso converges…

A: Just use definition of convergence.

Q: Let (an) be a Cauchy sequence. Suppose that for each H ∈ N, there is an m(H) ∈ N with m(H) > H…

A: To prove that limn→∞​an​=0 given the conditions on the Cauchy sequence (an​), we will use the…

Q: A) If (xn) is a decreasing and bounded sequence. Prove that (xn) is convergent with lim xn = inf {xn…

A: Multipart: (A)  We need to prove that , if xn is decreasing bounded sequence then , xn is convergent…

Q: Prove that the sequence {f(1/n)} has a limit if |f'(x)| < 1 for all r € (0, 1]. I

A:

Q: Let (an) be a sequence of real numbers that diverges to +∞, and let (bn) be bounded. Prove that lim…

A:

Q: Prove that limx → c f(x) = ∞ if and only if for every sequence an that is in the domain of f and…

A: As per the question we have to prove that limx→c f(x) = ∞ if and only if and only if for every…

Q: Prove from first principles that any Cauchy sequence (an) is bounded.

A:

Q: 6 If (rn) and (yn) are bounded real sequences, show that sup(n+yn) ≤ lim supan + lim sup yn lim 84x…

A:

Q: (d) (i) Define: sequence of real numbers {x} converges to the limit x = R. 3n+1 7n-4 (ii) Use the…

A: d.1. Definition:          A sequence of real numbers xn is said to be converges to the limit x∈ℝ…

Q: n=1 2m-1 m Let {} be the 'interlaced' sequence a, b, a, b defined by c₂ = a, c. = b Show that lim…

A:

Q: #6) Deler mine whether converges diverges, Calculatk limit an n+3

A:

Q: Let sequence (tn) = (1 , 0 , - 1 , 0 ,1 , 0 ,-1 ....) What is the lim sup (tn) , and lim inf (tn) ?

A:

Q: 6. (a) A sequence {anneN is said to be bounded if there exists M > 0 such that an M for any n E N…

A:

Q: If f is continuous and {an} is a sequence such that the limit lim an = L n➔ oo exists, then lim f…

A: Given: f is continuous at L.  an is an sequence such that limn→∞ an=L exists Now we have to prove…

Q: Find the limit of the sequence {an}, or show that it diverges: An 2n +1 In

A: Topic - limits