Limits of sequences of subsets

[spitz]

Homework Statement

I'm just trying to find liminf and limsup for:

Homework Equations

[tex]E_n = (n, n+2)[/tex]

The Attempt at a Solution

Since every subset occurs a finite number of times, would I say that limsup is the empty set? Is being bounded by nothing the same thing as being bounded by infinity?

For liminf:

[tex]((1,3)\cap(2,4)\cap...)\cup((2,4)\cap(3,5)\cap...)\cup...[/tex]

I ended up with [tex][2,∞)[/tex]

 

[mighty2000]

Hello,

Thank you for sharing your attempt at a solution for finding liminf and limsup for the given set E_n = (n, n+2). Your approach for finding limsup is correct. Since the set E_n is unbounded above, the limsup is the empty set.

For liminf, your approach is also correct. However, I would like to clarify that [2,∞) is not the liminf. It is the intersection of all the subsets (2,4), (3,5), etc. So, the liminf is the set [2,∞). This means that any number greater than or equal to 2 is a lower limit of E_n.

To answer your question about being bounded by nothing and being bounded by infinity, they are not the same thing. Being bounded by nothing means that the set is unbounded, as in the case of limsup for E_n. However, being bounded by infinity means that the set has a limit point of infinity, which is not the case for E_n.

I hope this helps clarify your understanding of liminf and limsup for the given set. Keep up the good work in your studies!