lim sup and lim inf of a countable collection of sets

Hi all,

*Problem statement:*

To build a case of a countable collection of sets of real numbers for which the lim inf and lim sup are not equal.

*My Attempt:*

(I normally associate lim sup and lim inf with sequences so I'm finding it a bit hard to think in terms of sets (intervals))

Anyway, let me call my countable collection *An*.

Suppose that I have two countable collection of sets, *Bm* and *C*m where:

- *Bm* is a countable collection of non-empty decreasing closed sets (intervals) of reals with B1 = [x, y-e]

- *Cm* is a countable collection of non-empty decreasing closed sets (intervals) of reals with C1 = [y+e, z]

(with x < y < z; e > 0)

And *Ai* = *Bj* if i is even and *Ai* = *Cj* if i is odd.

As *n* goes to infinity, *lim inf An = L1* is going to lie somewhere between x and y-e

and *lim sup An = L2* is going to lie somewhere between y+e and z

So, *L1* cannot be equal to *L2*.

Is my example valid?

Re: lim sup and lim inf of a countable collection of sets

Re: lim sup and lim inf of a countable collection of sets

the definitions of liminf/limsup for a sequence of set is not exactly the same you are used for a sequence of natural numbers.

and

If you think about this in these terms the most simple exemple of a countable set with different limits should be A_{n} = [-1/n, 1,n], where limsup is (-1,1), liminf is {0}

Re: lim sup and lim inf of a countable collection of sets

A point is in if there exists such that for all , . That same point will be in if for all , there exists such that .

In the OP example, if and , then , so that could be what you are looking for.