Let (an) be an increasing sequence of real numbers.Select all the true statements below.a.If there exists c ER such that an c for all n then (an) is convergent.□ b.If an #0 for all n then is decreasing.c.If there exists c ER such that an ≤ c for all n then every subsequence of (a) is convergent.

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Answered: Let (an) be an increasing sequence of…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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A. Show that any sequence of positive real numbers either has a subsequence that converges, or else a subsequence that diverges to ∞.If you could explain this a bit more detailed that would be perfect .
a-1 ak 1 an = 1 (a) For all ne N and a # 0, show that k=1 (b) For a sequence (xn) in R, suppose that there exists an b € (0, 1) such that |xn+1-Xn ≤ b for all n € N. Use part (a) to prove that xnx for some x € R. Note: If 0 1. Then use part (a), since a > 1 + 0.
If a
In which of the following options, the statements given below and the general terms an that satisfy these statements are matched correctly? I: is divergent when > an is convergent n=1 n=1 II: E- is divergent when >an is divergent n=1 an n=1 is convergent when the sequence {an} an n=1 III: is divergent K: an = n2 - Vn 1 an L: an M: = - 3 n (а) 1-М (b) I-L (c) I-M (d) I-L (е) I-К II - K II -M II - L II - K II - M III - L III -K III - K III - M III - L
Mark all the true statements about sequences of real numbers. Every sequence has a convergent subsequence. Every sequence has a monotonic subsequence. Every subsequence of a divergent sequence diverges. Every subsequence of a convergent sequence converges. Every sequence has a bounded subsequence.
Let x be a bounded sequence. Select ALL of the correct statements (note that there may be more than one correct statement). O None of these. O If X is decreasing then it is convergent. O X is convergent. O x has a monotone subsequence. O X has an increasing subsequence.
Let {n} and {n} be real sequences. Show that if xn → x and yn →y then for all a, ß ER, axn + Byn → ax + By

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