3. Prove that a sequence {a,} converges to the real number a if and only if lim sup a lim inf an = 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: Let (an) be a sequence and suppose that lim a2n = L = lim @2n+1. n→∞0 n→∞0 Prove that lim, an ² =

A:

Q: {xn} is a bounded sequence such that all convergent subsequences converge to the same number

A: Every bounded sequence has convergent subsequence.

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: Suppose that the sequence {n} is such that xn 20 for all n eN and that lim(-1)"xn exists. Show that…

A: We will use Triangle inequality.

Q: 1 Prove that if a sequence {a„} converges to a, then the sequence {[an]} converges to |a|. Find…

A: Suppose {an} converges to a. Prove that an converges to a. Also, check the converse is true.

Q: 5. Let {a,} be a sequence such that a, =1 and a =7. an-1 a) Prove by using mathematical induction…

A:

Q: Prove that if a sequence (an)1 is not bounded, then there is a subsequence (akn)-1 which either…

A:

Q: Prove that if a sequence  {an} converges, it must be bounded. Is the converse true as well

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: 81. Happy Thanksgiving!!

A: We have to deduce that an is convergent and find its limit.

Q: Let (an) be a bounded sequence. (a) Prove that the sequence defined by yn = sup{ak k ≥n} converges.…

A:

Q: Let {n} be a bounded sequence. Show that {n} converges iff lim sup (n) = lim inf (x₂)

A:

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

A:

Q: Give an example of a divergent sequence {a,} such that lim Ja,| converges. n 00

A:

Q: 2. Find the limit of the following sequences. a) {314

A:

Q: 6n Determine if the sequence an 4(-1) converges or diverges. If it converges, compute its limit. If…

A:

Q: Let {an}1 be a sequence. (a) Suppose that {an}1 is convergent. Prove that limn→∞ an+1 exists and lim…

A:

Q: 2. Let an be the sequence defined inductively by a₁ = 2 and an+1 (a) Prove by induction that an €…

A:

Q: Find a sequence of real numbers {a;} such that lim a, does not n-00 xist, but it has 3 subsequential…

A:

Q: 1. Let a, = 1 and an+1 = 1+ 1+an (a) Find the first six terms of the sequence {an}. (b) Prove that…

A: As you asked multiple sub parts so I answered 1st three subpart.

Q: Solve all 1) Assume that (xn) is a monotone decreasing and bounded sequence. Prove that (xn) is…

A: is monotone decreasing and bounded sequence.We have to prove that  is convergent.

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.