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Let (xn) be a bounded sequence. Denote yn=sup{xm : m is greater or equal to n}. Prove
(a) (yn) is a decreasing and bounded below.
Thus, by MCT, there exists a lim(yn)=inf(yn). this limit is called limsup(xn).
(b) Is (yn) necessarily subsequence of (xn)? if so, argue why, if not, give an example where (yn) is not a subsequence of (xn).
(c) Is it true that every bounded sequence (xn) has a subsequence convergent to limsup(xn)?. If so prove it (This would provide an alternative proof for BWT!)
I feel quite clueless- what elements are there in yn.
- how do you prove that yn is decreasing if xn is increasing?
-is xm a subsequence of xn?
-also it would help if I could get a rough idea of how to prove part a and c.
(a) (yn) is a decreasing and bounded below.
Thus, by MCT, there exists a lim(yn)=inf(yn). this limit is called limsup(xn).
(b) Is (yn) necessarily subsequence of (xn)? if so, argue why, if not, give an example where (yn) is not a subsequence of (xn).
(c) Is it true that every bounded sequence (xn) has a subsequence convergent to limsup(xn)?. If so prove it (This would provide an alternative proof for BWT!)
I feel quite clueless- what elements are there in yn.
- how do you prove that yn is decreasing if xn is increasing?
-is xm a subsequence of xn?
-also it would help if I could get a rough idea of how to prove part a and c.