Let {an}n≥1 be a convergent sequence of real numbers and let a ∈ R such that limn→∞ an > a. Show that there exist n0 ∈ N such that an > a for all n ≥ n0.

Q: Let the sequence (xn) defined by 1 X1 = 3, Xn+1 4-xn (a) Show that 0 < xn < 3 for all n E N (Hint:…

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Q: 1. Define when a sequence X = (x,) in R is said to converge to x E R. Then prove that the sequence X…

A: The given sequence is: xn=3n+12n+5

Q: 2. Let {an}, be a sequence of real numbers. Let a2k and a2k+1 be the even and odd terms of this…

A: To prove that if complementary subsequences converge to same limit then sequences also converges to…

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: Let a, be a dounded nonnegative sequence prove that: 1 lim sup n-0 (1+xn) 1 lim sup(1 +n) 1 lim inf…

A: Given, xn be a bounded nonnegative sequence. To prove that             lim supn→∞1(1+xn)=1lim…

Q: Prove that if {a,} is a convergent sequence, then to every posi- tive number ɛ there corresponds an…

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Q: True or false? d) Suppose (an) is Cauchy and that for every n ∈ N, the interval (−1/n,1/n) contains…

A: Given statements are : 1) Suppose (an) is Cauchy and that for every n ∈ N, the interval…

Q: 3) The space C is the set of all Convergent sequences in e. please Prove that C is complete.

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Q: Assume that the sequence converges and find its limit. a 1 = 7, an + 1 = 7an 7 -7 1 49

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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: Prove that following sequences converges to zero. 2n Xn n² + Tt

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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…

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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: Given (an) be a sequence of real numbers. Prove that if lim sup |an| = a for some a ∈ R, then (an)…

A: Limit superior: The greatest limit point of a bounded sequence is called limit superior. It is…

Q: 3. Prove that a sequence {a,} converges to the real number a if and only if lim sup a lim inf an =…

A: First suppose sequence  an is converges to a Then we have to prove that limsupan=liminfan=a Fix an…

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: 30

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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: Let (an) be a sequence of positive real numbers such that for all n, an+1/an > e^n . Use the…

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Q: 2. Every Cauchy sequence in the Euclidean metric space R, where n is a positive integer, is…

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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: (Bolzano-Weierstrass Property) Does every sequence of points in E have a subsequence that converges…

A: Consider the sequence

Q: Let A be a set of real numbers such that infla1a2a₁, 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: The statement…

Q: Let {an}n≥1 be a convergent sequence of real numbers. 1) Show that if for all but finitely many an…

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Q: Question * Every Cauchy sequence in the Euclidean metric space R" with n a positive integer is…

A: True. I have proved using definition of cauchy and convergent sequence.

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 Fn denote the Fibonnaci sequence defined by Fn = F(n-1) + F(n-2) for n >= 2 with initial values…

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Q: Examine the increasing and decreasing of each of the following sequences, whether they are bounded…

A: (i) The given sequence is an=n+11+3n To find: the given series is increasing or not, bounded or not,…

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: For each sequence an find a number k such that lim n"an is a finite non-zero number. Then use the…

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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: 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&gt;0 there exists K such that for n&gt;K,…

Q: 2. Every Cauchy sequence in the Euclidean metric space R", where n is a positive integer, is…

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Q: Let an = n 2 1 In(2" + 1) Determine if the sequence {an}n=1 is convergent or divergent 00

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Q: Let {an}, be a sequence of real numbers. Let a2k and a2k–1 be the even and odd terms of this…

A: We have to prove that sequence an converge to L.

Q: Let K be a nonempty compact subset of R. Prove every sequence in K has a convergent subsequence that…

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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: Exercise 10. Suppose {an}-1 be a sequence of positive real numbers and 0 < x < 1. If An+1 < x · an…

A: Let ann=1∞ be a sequence of positive real numbers and 0&lt;x&lt;1. an+1&lt;xan     for…

Q: Suppose that the sequence {an} is monotone. Prove that {an} converges if and only if {a%} converges.

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Q: 1. Suppose {Tn} is a sequence of real numbers that is (i) increasing (that is, In < In+1 for all n)…

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Q: Let {an} be a sequence of real numbers. Suppose that the subsequences {a2n} and {a2n-1} converge to…

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Q: 9. Let (x) and (y) be two sequences converges to c with x y for all n. Assume that f is defined for…

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Q: (a) Let {n} be a sequence such that there exist A> 0 and c € (0, 1) for which n+1-In ≤ Ac" for all…

A: It is given that, xn be a sequence such that there exist A&gt;0 and c∈(0, 1) for which xn+1-xn ≤ Acn…

Q: Prove if ( a n ) is a Cauchy Sequence and ( a n ) has a subsequence that converges to L, prove that…

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

A: To Prove: the sequence  yn=xn1+xn is convergent. Also, we need to give an idea of the limit of the…

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.