9. (i) For a real-valued sequence {an}, define precisely a) the notion of convergence to a finite limit L, and the notion of being Cauchy; b) show that convergence implies Cauchy; c) Does Cauchy imply convergence in R? d) Please, illustrate a context in which Cauchy does not imply convergence. (ii) prove that every convergent sequence is bounded, but the converse is false. Give an example of a sequence which is bounded but not convergent; (iii) prove that every Cauchy sequence is bounded, but the converse is false. Give an example of a sequence which is bounded but not Cauchy; . a) Show that Vk E N, Jan+k - an < 2/n ; b) does (iv) Let now a,n = Vn. the result of (i) -a) imply that the sequence {n} is Cauchy? (Explain and motivate clearly your answer)
9. (i) For a real-valued sequence {an}, define precisely a) the notion of convergence to a finite limit L, and the notion of being Cauchy; b) show that convergence implies Cauchy; c) Does Cauchy imply convergence in R? d) Please, illustrate a context in which Cauchy does not imply convergence. (ii) prove that every convergent sequence is bounded, but the converse is false. Give an example of a sequence which is bounded but not convergent; (iii) prove that every Cauchy sequence is bounded, but the converse is false. Give an example of a sequence which is bounded but not Cauchy; . a) Show that Vk E N, Jan+k - an < 2/n ; b) does (iv) Let now a,n = Vn. the result of (i) -a) imply that the sequence {n} is Cauchy? (Explain and motivate clearly your answer)
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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![9. (i) For a real-valued sequence {an}, define precisely a) the notion of
convergence to a finite limit L, and the notion of being Cauchy; b) show
that convergence implies Cauchy; c) Does Cauchy imply convergence in R?
d) Please, illustrate a context in which Cauchy does not imply convergence.
(ii) prove that every convergent sequence is bounded, but the converse is
false. Give an example of a sequence which is bounded but not convergent;
(iii) prove that every Cauchy sequence is bounded, but the converse is false.
Give an example of a sequence which is bounded but not Cauchy;
Vn.
a) Show that Vk E N, Jan+k – an] < ; b) does
(iv) Let now an
the result of (i) -a) imply that the sequence {/n} is Cauchy? (Explain and
motivate clearly your answer)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0a7d9fc5-182b-4619-bcf5-f19f3bccad39%2Fab441fa1-7ec0-407c-89b2-239552bff42b%2Fsc0pdev_processed.jpeg&w=3840&q=75)
Transcribed Image Text:9. (i) For a real-valued sequence {an}, define precisely a) the notion of
convergence to a finite limit L, and the notion of being Cauchy; b) show
that convergence implies Cauchy; c) Does Cauchy imply convergence in R?
d) Please, illustrate a context in which Cauchy does not imply convergence.
(ii) prove that every convergent sequence is bounded, but the converse is
false. Give an example of a sequence which is bounded but not convergent;
(iii) prove that every Cauchy sequence is bounded, but the converse is false.
Give an example of a sequence which is bounded but not Cauchy;
Vn.
a) Show that Vk E N, Jan+k – an] < ; b) does
(iv) Let now an
the result of (i) -a) imply that the sequence {/n} is Cauchy? (Explain and
motivate clearly your answer)
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