3. Let {an be a sequence in R. (a) Show that it has a monotonic subsequence. (b) Prove that a bounded sequence in R has a convergent subsequence. (c) For n E N, let an E [0, π) such that an = n mod T i. Show that {an}1 has a convergent subsequence. =1 ii. Is {an}1 dense in [0, π]? (Hint: Show that there exists a convergent subse- quence and divide [0, 7] into subintervals of length less than given ɛ.) iii. Is {n sin n n E N} dense in R?
3. Let {an be a sequence in R. (a) Show that it has a monotonic subsequence. (b) Prove that a bounded sequence in R has a convergent subsequence. (c) For n E N, let an E [0, π) such that an = n mod T i. Show that {an}1 has a convergent subsequence. =1 ii. Is {an}1 dense in [0, π]? (Hint: Show that there exists a convergent subse- quence and divide [0, 7] into subintervals of length less than given ɛ.) iii. Is {n sin n n E N} dense in R?
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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![3. Let {an1 be a sequence in R.
(a) Show that it has a monotonic subsequence.
(b) Prove that a bounded sequence in R has a convergent subsequence.
(c) For n & N, let an E [0, 7) such that
An = n
mod T
i. Show that {an}_1 has a convergent subsequence.
n= =1
ii. Is {an} dense in [0, 7]?(Hint: Show that there exists a convergent subse-
quence and divide [0, π] into subintervals of length less than given ɛ.)
iii. Is {n sinn n E N} dense in R?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffe72e13b-12e8-4646-917c-376e7356872c%2F5e93421e-1707-4d76-862e-fc32ca98dffe%2Fv0wkwe_processed.jpeg&w=3840&q=75)
Transcribed Image Text:3. Let {an1 be a sequence in R.
(a) Show that it has a monotonic subsequence.
(b) Prove that a bounded sequence in R has a convergent subsequence.
(c) For n & N, let an E [0, 7) such that
An = n
mod T
i. Show that {an}_1 has a convergent subsequence.
n= =1
ii. Is {an} dense in [0, 7]?(Hint: Show that there exists a convergent subse-
quence and divide [0, π] into subintervals of length less than given ɛ.)
iii. Is {n sinn n E N} dense in R?
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