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 reasonabledefinition for lim inf an and briefly explain why it always exists forany bounded sequence.

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: 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: 1. Let (an) be a bounded real sequence. Let and U = {u: (³N)(\n > N)(an N) (an> l)}. Prove that sup…

A: As per the question we are given a sequence (an) and based on that sequence two set of limiting…

Q: Let (sn) be a sequence with lim sup sn = 10 and lim inf sn = 5. (a) Show that (sn) is bounded. (b)…

A: Consider a sequence, Sn, with lim sup Sn = 10 and lim inf Sn = 5. (a) To show that Sn is bounded. By…

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: Prove: Suppose that lim an 46. Prove n→∞ that the sequence {an} is bounded below. =

A:

Q: 3] Let (xn) be bounded (not necessarily convergent) sequence, and assume that lim(yn) = 0. Show that…

A: A sequence is said to be bounded if it is bounded above and below by a number. That is, the sequence…

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

A: Let f is continuous at c and xn be a sequence such that limnxn=c Since, f is continuous at c, for…

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: Consider the following sequence: en + en (a) Show that the sequence is monotone and bounded (and…

A: Find the limit of the sequence

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

A:

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: 2. Let an be the sequence defined inductively by ai = 2 and an+1 = 1 an + 2 an (a) Prove by…

A: Disclaimer: Since you have posted a question with multiple sub-parts, we will solve first three sub…

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…

Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence.  It was invented by German mathematician Carl Friedrich Gauss.