Thread: Limit Superior Definition Proof

Maybe someone can help me out, I thought I had an idea but I don't think it is right.



I am trying to prove the above. My attempt:

I am given that the lim sup Xn is finite, which I think means it exists, which I will call L. So I have a sequence [Xn] = [x1, x2, x3,... xn, xn+1,....] and another sequence [Xm] = [xn, xn+1,...]. This is where I am stuck, I don't know where the next step should be. I have an idea that if I can show [Xm] is monotonically decreasing then the sup of [Xm] is Xn and then taking the limit of Xn gives me L, which would complete the proof, I think. But, I don't know how to show that the sequence is monotonically decreasing, or if it even is... it was just an idea. Any help/hints would be appreciated!

Originally Posted by Math150

Maybe someone can help me out, I thought I had an idea but I don't think it is right.

I am trying to prove the above.

How does one prove a definition?
What definition of have you been given if it is otherwise.

Thanks for replying Plato, and I know it seems weird to prove a def, I guess I mean to just show that it is true. I've defined lim sup to be the supremum of the set of subsequential limit points of the sequence, {Xn}... The problem lets me assume that the set is finite, so I've been calling the sup of the set, L.


So the set of subsequential limits = [..., L] I have to show somehow that the lim sup Xm (m>=n) is also L.

I read somewhere that if I have a sequence [An] then I can define a new sequence [Bn] = sup Ak, k>=m = sup [An, An+1, An+2,...] I am trying to incorporate this and get something out of it because then the sequence Bn is montone decreasing

Okay the last post is getting cluttered, I am trying to work on a new solution. I have:

Given a sequence Sn = {S1, S2, S3, ... Sn, Sn+1,...}
Let E be the set of subsequential limits of Sn. Assume E is finite then E= {....L}
This means that the Lim sup Sn = L.

Now for the right half of the equation. Set Tn = sup Sk , k>=n. Then by substitution we have Lim Tn on the right.

So as of now it is:

L = Lim Tn (n goes to infinity) ... I just need to show this somehow.

L = Lim sup (Sn, Sn+1,...)

I know that Tn is a decreasing and bounded sequence so it does converge, but I just don't know how to show it is equal to L or the limit superior of Sn.

Originally Posted by Math150

Given a sequence Sn = {S1, S2, S3, ... Sn, Sn+1,...}
Let E be the set of subsequential limits of Sn. Assume E is finite then E= {....L}
This means that the Lim sup Xn = L.

I really am not sure of what you are about here.
But any finite set contains a maximum term.
If is the set of subsequential limits of and is finite then

Okay, I thought it was the last term, but if it's maximum I will just reword it... All I meant to do was denote the maximum as L.

also that last line in quoted text should be Sn, not Xn.


Can I say as n is going to infinity, n must be getting closer and closer to k... so at some point I will have that sup Sn -> Sup Sk which would be lim (Tn) => lim sup (Sk) => lim sup (Sn) => L. ... On second thought this might not make much sense.


I just need to find a way to show that the decreasing and bounded Tn satisfies:

Lim Tn as n-> infinity = L

Where Tn is sup Sk for k>=n

Just thinking... since Tn is decreasing and bounded then the Lim of Tn must be = inf Tn.

Can I say that the existence of the limit superior means that Sn is bounded above, L. If I define Tn as sup Sk k>=n then:

T1 = sup {S1, S2, S3, S4,...}
T2 = sup {S2, S3, S4, ...}
T3 = sup {S3, S4,...}

This shows that each one is a subset of the one before and thus are decreasing. So: T1>T2>T3>... Tk-1>Tk>... Each T is bounded below and above (by the original Sn) So the Limit of Tn is then the same as the upper bound, L. Does this make sense?

Originally Posted by Plato

How does one prove a definition?
What definition of have you been given if it is otherwise.

I have been asked to prove the same thing... I too do not understand WHY we have been asked to prove a definition but there it is! I suppose it's sufficient to show that since E is bounded above (by s*=lim sup(xn)), if any subsequence were to have a limit (sup xm) greater than the upper bound of E, that would contradict the definition of E and its boundedness? I'm not coming up with any better ideas!