Proving the Convergence of Subsequential Limits: Solving for lim sup and lim inf

[Conlan2218]

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 s_n is an element of SL(s_n)
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 s_n is an element of the set of all limits of all convergent subsequences of the sequence s_n. 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.

 

[jgens]

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.

 

[Conlan2218]

I am having trouble showing that there must be a subsequence that converges to the sup any suggestions?