Royden Review:LimSup, LimInf are the Largest/Smallest Limit Points of {a_n}

[Bacle]

Hi, All:

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.

 

[HallsofIvy]

What is your definition of "limsup" and "liminf". The definition I learned was "[/itex]a= limsup a_n[/itex] if and only if a is the supremum of the set of all subsequential limits of $a_n$" and "$b= liminf a_n$ if and only if b is the infimum of the set of all subsequential limits of $a_n$".

But notice that I used "sup" and "inf", NOT "largest" and "smallest". A sequence may not have "largest and smallest limit points".

 

[micromass]

Let's say that L is a limit point of the sequence. Can you prove that

$$\sup_{n\geq k}~{a_n}\geq L$$

for every k?