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Give an example of i) a Cauchy sequence which is not monotonic; ii) a monotonic sequence which is not Cauchy; iii) a bounded sequence which is not Cauchy; iv) Can you find an example of a Cauchy sequence which is unbounded? If not, why not? Explain your answers.

Give an example of i) a Cauchy sequence which is not monotonic; ii) a monotonic sequence which is not Cauchy; iii) a bounded sequence which is not Cauchy; iv) Can you find an example of a Cauchy sequence which is unbounded? If not, why not? Explain your answers.

Advanced Engineering Mathematics
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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Give an example of i) a Cauchy sequence which is not monotonic; ii) a monotonic sequence which is not Cauchy; iii) a bounded sequence which is not Cauchy; iv) Can you find an example of a Cauchy sequence which is unbounded? If not, why not? Explain your answers. For (i) take = xn= Show that {xn} is a Cauchy but not monotonic. For(ii), take= xn=n Show that{xn} is monotonic. Take &=;n E N; and n > N and m=n+1 Explain why (-1)" n |xn−Xn+1| > E Conclude that {xn) is monotonic but it is not Cauchy. For (iii), take xn = 1 +(-1)^. Show that {xn} is bounded, i.e. show that |xn| <2; for all n E N. Compute |xn-Xn+1| Explain why {xn} cannot be a Cauchy sequence. For (iv) Recall that every Cauchy sequence is bounded. Conclude that no such an example of a sequence {x} can be constructed.
Transcribed Image Text:Give an example of i) a Cauchy sequence which is not monotonic; ii) a monotonic sequence which is not Cauchy; iii) a bounded sequence which is not Cauchy; iv) Can you find an example of a Cauchy sequence which is unbounded? If not, why not? Explain your answers. For (i) take = xn= Show that {xn} is a Cauchy but not monotonic. For(ii), take= xn=n Show that{xn} is monotonic. Take &=;n E N; and n > N and m=n+1 Explain why (-1)" n |xn−Xn+1| > E Conclude that {xn) is monotonic but it is not Cauchy. For (iii), take xn = 1 +(-1)^. Show that {xn} is bounded, i.e. show that |xn| <2; for all n E N. Compute |xn-Xn+1| Explain why {xn} cannot be a Cauchy sequence. For (iv) Recall that every Cauchy sequence is bounded. Conclude that no such an example of a sequence {x} can be constructed.
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