true or false , prove your answer If (an) is Cauchy and p is a number such that for all k ∈ N, |ak − p| < 1/2, then (an) converges to a limit L that lies in the interval [ p − 1/2, p + 1/2 ].

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A: We use the definition of Cauchy sequence to prove part f

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Q: 1. Prove the sequence (a,} = (1+" Converges to the limit e*, what about the sequence {b,} = (1-5?…

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

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Q: Prove the limit comparison theorem i. Suppose that an20 and b,20 and that an = lim n+ 00 Then either…

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Q: ii) Prove that if a sequence XnXn converges to a limit l, then any subsequence of XnXnalso converges…

A: Just use definition of convergence.

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A: We can prove it by showing that function is not pointwisw convergent.

Q: true or false prove your answer If (an) is Cauchy and p is a number such that for infinitely many…

A: Given that              an is a Cauchy sequence and p is a number such that for infinitely many…

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Q: 5. True or False? You do not need to prove your answers (but know how you would prove them!). e) If…

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Q: Prove that the sequence is monotone and bounded. Then find its limit. s1 = 5 and sn+1 = V4sn +1 for…

A: See the attachment

Q: 1. Prove the following lemma: 8. Lemma A. Let {xn}1 be a sequence of real numbers. We define two new…

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Q: 1 then (an) converges to 0. An+1 Exercise 8. Prove that if (xn) is a sequence such that lim n→∞ An…

A: Given that, 8. The sequence is limn→∞an+1an=12 then an converges to 0. 9. The sequence is…

Q: a) If lim an = 0, then Σ an converges. n→1

A: The divergence test states that for a series ∑n=1∞an, if limn→∞an ≠ 0 , then ∑n=1∞an diverges . That…

Q: Consider the sequence of functions {fn}nen, where nx fn(2)= Væ€ [1, 10]. 1+nx (a) Find the pointwise…

A: Given :  fn is a sequence defined by ;  fn(x) = nx1+nx ,  ∀ x∈[1, 10]. To find :  (a) The pointwise…

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Q: Suppose that {an} converges to a and lal < 1. Prove that {(an)"} converges to 0. (NOTE: You may not…

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Q: 1. Let Cn+1 = V2/Cn. show {cn} converges to c. Find c. Hint: Use induction to show cn is bounded and…

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Q: Find the mistake in the following “proof": "Statement". If fn : X → Y is a sequence of functions…

A: Given: Let us consider the given, ∈&gt;0 and uniform convergence, and determine N∈ℕ.

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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.