Let {an}n≥1 be a convergent sequence of real numbers. 1) Show that if for all but finitely many an we have an ≥ a, then limn→∞ an ≥ a. 2) Show that if for all but finitely many an we have an ≤ b, then limn→∞ an ≤ b. 3) Conclude that if all but finitely many an belong to the interval [a, b], then limn→∞ an ∈ [a, b].

Q: Prove that if the sequence {an} is divergent and the sequence {bn} is convergent, then the sequence…

A: Given: The objective is to prove that if the sequence {an} is divergent and the seqence {bn} is…

Q: Q4] Show that the sequence (xn) = (T") for all n EN is not a Cauchy .sequence

A: A sequence {Xn}of real numbers is called a Cauchy sequence if for every positive real number ε,…

Q: Prove that the sequence (xn) = (-1)"+1 [3 .does not converge to any real number

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Q: (a) If (an) is a sequence such that (an bn) converges to zero for every sequence (bn) converging to…

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Q: If g : A → R is continuous and (an) is a Cauchy sequence in A, does it follow that the sequence…

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Q: Show that the sequence defined by 1 a₁ = 1 an is increasing and an < 3 for all n. Deduce that {an}…

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Q: Suppose an is conditionally convergent. Define what this means, and then use a proof by n=1…

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Q: Let [[x]] be the greatest integer less than or equal to x. Forexample, [[π]] = 3 and [[3]] = 3. For…

A: We have given; [[x]] be the greatest integer less than or equal to x.

Q: Suppose a Cauchy Sequence {xn} is such that for every M ∈ N, there exists a k ≥ M and n ≥ M such…

A:

Q: Suppose a Cauchy sequence {xn} is such that for every M ∈ N, there exists a k ≥ M and an n ≥ M such…

A: every cauchy sequence of real numbers is convergent. And a sequence can have at most one limit.

Q: Let an be an infinite sequence of real numbers. If an converges, then >-1 an converges. 'n=1

A: Consider the provided question, We have to check the given statement is true or false, Part first:

Q: Let E ⊆ (Real Number) and E does not equal ∅.  Prove if E is compact then every sequence in E has a…

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Q: 7. a) Let x = c, with ||- Give an example of a Cauchy sequence in X that do not converge in X.…

A: Here, X=C00  is set of all sequence having only finitely many non zero terms with the norm xp where…

Q: 30

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

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Q: 1. Let {an}1 be a convergent sequence such that V5 and an+1 3D V5а,. a1 = Compute lim an. n→∞

A: ann=1∞ be a convergent sequence such that a1=5 and an+1=5an

Q: A Cauchy sequence f} in L₂ converges to an element f in L₂

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Q: (a) Prove that if > ak converges absolutely and (bk) is a subsequence of (ar) then ) bk also k=1 k=1…

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Q: 6. Suppose a, converges. Assume the terms a, are non-zero. Prove ÷ diverges if it is true. n=1 n=1…

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Q: If {an} and {bn}1 are sequences of real numbers such that 0 < an < bn for all n 1and Σ An n=1…

A: A detailed solution is given in Steps 2 and 3.

Q: 3. Determine if the sequence xn = 47² is convergent or not and find its limit if it exists n²+1

A: We want to find the given sequence is convergent or divergent.

Q: Suppose (an) is Cauchy and that for every k ∈ N, the interval (−1/k, 1/k) contains at least one term…

A: Given, an is Cauchy sequence, Let, ank∈-1k,1k ,∀k∈ℕ Then, ank1k , ∀k∈ℕ Let, ε&gt;0 be arbitrary and…

Q: . Let (xn)neN be a sequence in R. For each n in N, let X1 +x2 + +Xn Yn Show that if (xm)nɛN…

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Q: Let (an) be a sequence in which every term is from some given finite set R. Prove that (an) has a…

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Q: (a) Prove that, for each k E N, b - Ck = ak and b+ Ck = |ak]. (b) Prove that, if ak converges…

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Q: Every Cauchy sequence in the Euclidean metric space R" with n a positive integer is convergent. O…

A: Cauchy sequence and convergence

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Q: 1. Let (n) be an increasing sequence of positive real numbers and define the sequence (yn) as…

A: Given that xn be an increasing sequence of positive real numbers and define the sequence yn as…

Q: 4. Let A be a set of real numbers such that inf{a1a2a1, a2 € A a1 # a2} > 0. Prove that every Cauchy…

A: Given :  A be a set of real numbers such that ;   infa1-a2 : a1, a2 ∈A, a1≠a2 &gt; 0 To prove :…

Q: @ suppose {an} and { bn} are sequences such that {an} and { ant bn'} converge.prove that {bn}…

A: To prove (a)  Given that an converge and an+bn converge. The objective is to show that bn converge.…

Q: If H be a Hilbert space and C be a subset of H. Give an example of a function, T:C- C, which…

A: Given that H is a Hilbert space and C is a subset of H. To find  a function that satisfies the…

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

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Q: 20

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Q: 4. If {Cn} is a convergent sequence of real numbers, does there necessarily exist R > 0 such that…

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Q: Give an example of two sequences (an) and (bn) such that all of the following are simultaneously…

A: To Give: An example of two sequences an and bn such that all of the following are simultaneously…

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Q: 1. Consider the sequence {an}, where a, e [a, b] for all n e N. Show that is convergent.

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Q: 2. a) Show that for any positive integer k, the sequence defined by pn = converges linearly to p =…

A: Since you have asked multiple questions, we will solve first question for you. If you want other…

Q: Suppose (an) is Cauchy and that for every k ∈ N, the interval (−1/k, 1/k) contains at least one term…

A: Observe that the given condition says that 0 is a subsequential  limit of the sequence.

Q: Prove that any convergent sequence {xn}=1 is bounded.

A: We want to prove every convergent sequence is bounded.

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Q: 14. Suppose (an) is a bounded sequence such that all of its converging subsequences converge to the…

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Q: Suppose {a,} is monotonic. Prove that {an} converges if and only if {a} converges

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Q: Suppose that (an) and (bn) are convergent sequences of real numbers with limits an → a and bn → b,…

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Q: If sequences {an} and {bn} both diverges to infinity, An is convergent. bn then the sequence

A: We have to find

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.