Let (n) be a bounded sequence and let s = sup{r, ne N}. Show that if se {an:ne N), then there is a subsequence of (a) that converges to s.

Q: Determine the monotonicity and boundedness of the following sequence 4" 2n + 100 an=

A: Determine the monotonicity and boundedness of the following sequence a_{n} = (4 ^ n)/(2 ^ n + 100)

Q: Let x₁ = √2, and x1 Xn+1 = 2 + √√xn, n = 1, 2, 3, .... Prove that the sequence {n} converges, and…

A: Let  and To Find:Prove that the sequence is monotonically increasing and bounded above also shows…

Q: n 11. Let X be a non-empty set that is bounded from below. Show that there is a decreasing sequence…

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Q: 5. The subsequence of (xn) = () is.. a) (1층..) 0) (1.층,급.) b) (1,1,1, ..) a) (1.-) 6. The sequence…

A: Every subsequence has a pattern.  According to guidelince we solve only first question.

Q: see the question as attached here

A: given,       an diverges

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: Let (an) be a bounded sequence. Define (b,) by bn := sup{an, an+1, An+2;...}. Prove that (bn)…

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Q: A sequence an converges in the mean if an n 1a converges. Prove that if an → L, then ān → L, but the…

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Q: Suppose Sanx",Σ bnan converges for all x in (-1,1). Σana" = Σbnan in a subset E of (-1,1) implies…

A: It is given that,  ∑anxn and ∑bnxn are convergent for all x in (-1, 1). Then,  ∑anxn = ∑bnxn in a…

Q: If {f} converges in the mean squa

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Q: Let p be a given Natural Number. Give a sequence (XN) that is not a Cauchy sequence, but it…

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Q: EXERCICE 7. Let be the recurrent positive sequence (аn)2 3 1 + Зan-1 avec ao = 0 a) Provide the…

A: As per our guidelines we are supposed to answer only one question.Kindly repost other question as…

Q: Choose the correct answer. 1. The sequence (x) = (-n) is. a) Cauchy b) bounded e) divergent d)…

A: We have1) The sequence (xn)=(-n) is . . . . . . . . .Now we have take n=1 x1=-1put n=2x2=(-2) So, we…

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

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Q: Q3) The sequence a, = 1 - (0.2)" a) converges to 0 b) diverges c) converges to e d) converges to 1

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Q: 1. Prove that the sequence {4+} converges to 4.

A: To know weather two function converge or not take difference of them and put limit to infinite. If…

Q: Suppose real sequences (sn) and (tn) are bounded. (That is, that their ranges are bounded sets.) i.…

A: Bounded sequence and sum of bounded sequences.

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

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Q: == 2.9 Let (an) be a sequence with limit a, and define bɲ = an+1 (n = 1, 2,...). Show that (bn) → a.

A: We know that  A sequence an converges to a limit L  if, for any positive real number epsilon there…

Q: Let be a sequence of functions belonging to L P-space. If this seque is n onvergent, then it is a…

A: Sol:- The statement is true.

Q: In be a sequence satisfying 0 < In <1 and 4.xn+1(1-rn) 21 for all n E N. ow that {In} coverges to…

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Q: If (xn) is a Cuachy sequence in R, then .it is unbounded :Select one True False

A: We Know that A sequence {n}is called a Cauchy sequence if for any given ϵ &gt; 0, there exists N ∈ N…

Q: Let (an) be a bounded real sequence, and denote by anfn the product of an and fn. Show that (anfn)…

A: We have to show that the sequence () has a subsequence that converges uniformly on a set A. We can…

Q: Let {n} be a Cauchy sequence. Prove that {s} is bounded.

A: Given: In this question, the given details are, Let {sn} be a Cauchy sequence. To Prove that {sn} is…

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

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Q: 10. Which of the following statements is /are not true? I. Every convergent sequence is Cauchy. II.…

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Q: ii) xn √n-1 √n+1' iii) x₂ = (-1)"+¹ + 1/1.

A: We recall cauchy criterion and then solve this question. Also needed will be the Archimedean…

Q: 2. Let (an)1 be a sequence of integers. Prove that (an) converges if and only if there exists a…

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Q: 2. Prove the sequence an = n"/n!, n > 1, is %3D (a) increasing; (b) not bounded above (show an 2 n).

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Q: The sequence {G 3n+2 n=1 O None converges to e O diverges converges to Ve? converges to e3

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Q: 5. Use the definition of convergence to prove that the sequence (sn) diverges where Sn = √n.

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