- #1

Bacle

- 662

- 1

Let {a_n}; n=1,2,... be sequence. I am trying to show LimSup and LimInf are the largest and smallest limit points of {a_n}. This is what I got so far:

i) If {a_n} converges, to, say, a<oo, then LimSup=LimInf, and we're done, since we have a unique limit point L. If a=oo, then oo is the limit point.

I think I can show ( here in ii) below ) that Lim Sup, Lim Inf are both limit points of {a_n}, but I cannot show they are the largest, smallest respectively.

LimSup is a limit point of {a_n} if {a_n} does not converge:

Proof:

ii)If {a_n} does not converge, then it is not strictly monotone, so we can extract monotone

non-increasing and monotone non-decreasing subsequences ( by using, e.g., the lim sup

and lim ii) lim sup, lim inf are both limit points; in the case of lim sup, the sequence: a_n' :

{ sup_k>n(a_n)} is monotone non-increasing; by the LUB property,L= LimSup{a_n} :=inf_n

(a_n') exists, and it is a limit point of {a_n}; by contradiction, if L were not a limit point of

{a_n}, there would be e>0 with L> a_k-e for all a_k. But then L is not the lub of the

monotone-decreasing subsequence; a_k-e is the LUB.

I am stuck trying to show that Lim Sup is the largest limit point; I am sure the proof that

Lim Inf is the smallest limit point is automatic after knowing this one.

Thanks.