An exercise about limsup/liminf of a subset sequence

Let with n natural number.

Calculate: E', E'' where E' = liminf E_{n}, E'' = limsup E_{n} for n that goes to

Looking at the goniomethrical discus i should say that liminf is the empty set and that limsup is \ , but how could i formalize it (if is correct) ?

Re: An exercise about limsup/liminf of a subset sequence

What is the infimum of ? Of ? Of ? Given any , choose any . Then . So, the infimum of is for all . Hence, .

The suprema are a little different. How do you find the supremum of ? We need only look at the interval . We know on the interval and on . So, the supremum of would be . So, what is ?

Re: An exercise about limsup/liminf of a subset sequence

the first answer should be and the other subsets depending on the periodicity so with the limitation that 2k+1/n < 2, 2k/n > 0 with k positive integer.

the second answer is looking at the sign of the prime derivative of sin(nx)/n.

the third answer should be . So the limsup should be { }

P.S. When you say 0 for E' you mean the subset {0} i suppose, anyhow thanks a lot for your help. I was struggling about this one.

Re: An exercise about limsup/liminf of a subset sequence

You seem a little confused about what an infimum or supremum is. It is a greatest lower bound or least upper bound respectively. You had an example before where you were looking at a set of sets. A set of sets can be viewed as a poset where the partial order is containment. if . In that context, an infimum or supremum will be a set.

In the context of the current problem, you do not have a set of sets. You have a set of points of the real line. In this context, a greatest lower bound or a least upper bound will be a point, not a set. So, the first answer is not the set . How do we know that is the infimum of the set? We know it is a lower bound since for all . Given any , which shows that is not a lower bound. Hence, 0 is the greatest lower bound, and therefore the infimum for each . So, is an infinite sequence of zeros. The limit of an infinite sequence of zeros is zero. That is what is meant by the limit inferior ( ). So, when I say 0 for E', I mean 0, not {0}.

Similarly for the supremum.

Re: An exercise about limsup/liminf of a subset sequence

Looking at my book of Mathematical Analysis III i found this definition: so is a set, not a value.

I thought you was talking about when you said infimum in this case.

Re: An exercise about limsup/liminf of a subset sequence

Anyhow if i expressed myself bad what the exercise asks is to calculate E', E'' defined as for

P.S. i'm sorry for the double post but doesn't let me edit the previous one.

Re: An exercise about limsup/liminf of a subset sequence

Using that definition, you are probably correct about .

Here is one possible approach to a formal proof:

Obviously, , so those points cannot be in or .

If , then for all . Given , suppose that for every , there exists with . Then, obviously for any . That obviously implies . So, if you can show that given any and any , there exists with , then you have proven .

For , if for all , there exists with , then . (This condition is dual to in a similar way to how intersections are dual to unions).

If you need any more help setting up the argument, I have some ideas.

Re: An exercise about limsup/liminf of a subset sequence