For each of the following, give an example of a sequence with that property and justify why it does so, or else prove that no such sequence exists: (a) A sequence that has a subsequence converging to 7, a subsequence converging to 17, and a subsequence converging to T. (b) A convergent sequence with an increasing subsequence and a decreasing subsequence. (c) A sequence with no convergent subsequences.

Q: Let (Sn) be convergent sequence, and suppose lim s, > a. Prove there exists a number N such that n>…

A: Given sn is a convergent sequence, and suppose limn→∞sn>a. Since sn is a convergent sequence,…

Q: Give a counterexample and explain why this statement is false: “If a sequence contains both a…

A: The given statement statement “If a sequence contains both a strictly increasing subsequence and a…

Q: Prove or Disprove: Every oscillating sequence diverges.

A: To investigate the convergence (or divergence) of an oscillating sequence.

Q: Show that the sequence defined by  a sub 1= 1  and  a sub n + 1 = 3 - (1/ a sub n) is increasing and…

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Q: Define the sequence {xn} by (see photo). Prove, using the epsilon-N definition, that this sequence…

A:

Q: If the sequence {a}, is convergent, then the sequence {x} defined by a₁+as+as++an-stan is also…

A:

Q: If ( a n) is a sequence that diverges to ∞, prove that Iim n --> ∞ (1/a n) = 0

A: Assume that the sequence an diverges to ∞. That implies, limn→∞an=∞.

Q: If (an) is a sequence converging to L, prove that (an)=1 converges to L.

A:

Q: (d) A Cauchy sequence with an unbounded subsequence. (e) An unbounded seuqence with a Cauchy…

A: The given problem is related with Cauchy sequence. We have to give an example of a sequence with the…

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: Every convergent sequence is bounded. O True O False

A: Every convergent sequence is a bounded sequence, that is the set {xn : n ∈ N} is bounded. Remark :…

Q: Let {an}1 be a sequence such that there exist numbers a and N such that, for n > N, an = a. Prove…

A: We know that the sequence an converges to α when for any ε>0 there exists K such that for n>K,…

Q: Define a sequence (an) recursively by setting a1 = V2 and an+1 = v2+ an for n > 1. (a) Show that an…

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Q: Let {a,} be an infinite sequence with the property that 0 < a, < for all n 2 1. Calculate lim,-00…

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Q: Every increasing monotonic and sequence is convergent

A: given the sequence is increasing monotonic

Q: Give an example of each of the following, or indicate that no such example example exists by writing…

A: An example for a convergent sequence that is not bounded. Let A=an be a convergent sequence. Then ,…

Q: List the first 5 terms of the following sequence and determine whether or not it 4n2-4 converges: an…

A:

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.