1. Suppose {an} and {bn} are positive sequences and that there exists N so that n >N = an < bn. Show that if£ bn converges, then so does an.

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: B) Prove that the sequence is convergent, where az = v2, = v2, 2+an, an+1 = %3D n 2

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Q: Define a sequence (Sn)n inductively by letting S1 = 1, and Sn+1 = Sn + 1. 1+V5 Prove that lim Sn = .…

A: Limit of a sn

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 {cn} converges to c if and only if the sequence {cn- c} converges to 0.

A: To prove that the relation between the limits of the two given sequences

Q: Determine the limit of the sequence a, = [In(n + 1)-in(n)] 1 e 1 in 2 1 4 e Divergent N O

A: Topic = Limits

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

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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: Let {an}n=1 be a convergent sequence such that a1= V5 and an+1= 5an Compute lim an

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Q: Consider the sequences (an) and (bn) given by an= (−1)nn2, and bn= 2n. Decide if the sequence…

A: Convergency for an:we can find limitwe know that (-1)^n is oscillating This value can be either -1…

Q: For the sequence {an}o with lim an exists, this means the sequence converges. Select one: O True O…

A: First is true and second is false.

Q: If (an) is a sequence converging to L, with an 20 for all n, prove that L≥ 0 and that (√an)1…

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Q: Prove that there is no hooked Skolem sequence of order n= 8.

A: To Prove - Prove that there is no hooked Skolem sequence of order n = 8.         Theorem used - A…

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: Suppose anconverges to 8 and bn converges to 5. What does (3a,- 2b,) converge n=1 n=1 n=1 to? Do the…

A: Please find the answer.

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: En3D1 In(n) 2 +1

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Q: Let (an), (bn) be sequences, so that (b,) is defined as b, = an – an+1· Prove that the series b, n=1…

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Q: Ean converges and an > 0 for all n , can anything be said about E? An

A: If ∑ an converges and an &gt; 0 for all n then ∑ 1an  converges         Proof :…

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

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

Q: In a metric space (S, d) every convergent sequence is a Cauchy sequence. Is the statement True or…

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

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Q: Suppose a,>0 for all n and 2 a, converges , which of the following must converge n = 1

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

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Q: Write out first five terms of the sequences, determine whether, the sequence converges, and if so,…

A: We have to find the limit of the sequence and find first five terms

Q: If an and bn are sequences such that 0 < an < bn and bn converges, then an converges. Select one: O…

A: We will solve this question using a example.

Q: Suppose that {a, } and {b,} are two positive sequences, and define up . u /2n² + bn %3D If lim (a,)…

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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: 10. Suppose {an} is a sequence of real numbers and b, the sequence {bn} is convergent, so is {an}. =…

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Q: Suppose that the sequence {an} converges to a   and that a&gt;0. Show that there is an index N such…

A: Prove the following as follows.

Q: Let {an}1 and {bn}n=1 be sequences. Prove or disprove: If {an – bn}=1 converges to 0 and {an}-1…

A: Note that, if limn→∞an and limn→∞bn exist, then limn→∞an-bn=limn→∞an-limn→∞bn. It is given that,…

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

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Q: 8. Suppose that for sequence of real numbers {an} and {b,} that O< bn+1<an < bn, Vn EN Prove that…

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Q: 2.21, Suppose that {an}n€N is a Cauchy sequence in a metric space X, d suppose there exists a…

A: We need to show xn converges to x

Q: Let ₁1 and In+1 = limit. 1 for n N. Show that this sequence is convergent and find its In + 2

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Q: determine the limit of the sequence or show that the sequence diverges.

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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. Use the definition to prove that if (xn) is a sequence of real numbers which converges to -3,…

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Q: Determine the convergence of the following sequences. {sech n} a.

A: We have to solve given Questions.

Q: Show that (1/n²+ n+1)!nEN is a Cauchy sequence.

A: Solution :-    The given sequence is              an =   1/( n2 + n + 1 )     ;  n belong to N .…

Q: If possible, give an example of 2 convergent sequences an and bn such that E(an + bn) diverges.

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Q: 3. Suppose that (z,) is a bounded sequence and let T= (t€R:1<I, for infinitely many terms z,). Prove…

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

Q: Show that the fractional parts of a given sequence {annz1 is uniformly distributed if and only if 1…

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Q: If (x,) is a sequence in R such that the series 2(x is convergent, then the series 2 (x) n=1 must be…

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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: an = 0 and bn Σ-1 bp Show that if lim n converges then En-1an converges too.

A: Given that:            limn→∞anbn=0 and ∑n=1∞bn converges. Prove ∑n=1∞an converges.

Q: of all sequences that are "eventually zero," that is, al

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Q: Let {X,} be an independent Bernoulli sequence with parameter p, and let np, A < o as n oo. Verify…

A: Verifying that the Lindeberg condition fails for Xn