Analysis subsequential limits

  • Thread starter Conlan2218
  • Start date
  • #1

Homework Statement



I missed the last class and I guess our teacher solved this problem and said it would be on our next quiz and I do not really understand how to do it.

Prove that

(i) lim sup sn is an element of SL(sn)
n to infinity

(ii) same thing but replace the sup with inf

Homework Equations









Homework Equations



lim sup:= limit supremum
lim inf:= limit infimum

The Attempt at a Solution




I know that it wants me to prove that the limit supremum of a sn is an element of the set of all limits of all convergent subsequences of the sequence sn. It makes sense that the sup and inf of the sequence would be would be elements of the set of all convergent subsequences because in order to have convergent subsequences the original sequence must be bounded. Thus, bounds would seem to be elements of the convergent, bounded subsequences.
 
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Answers and Replies

  • #2
jgens
Gold Member
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Why don't you try constructing a subsequence which converges to the limit supremum of a bounded sequence? You might need to consider a couple of cases, but it seems like it should be pretty straight forward.
 
  • #3
I am having trouble showing that there must be a subsequence that converges to the sup any suggestions?
 

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