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

Quote:

Originally Posted by

**director** 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?

I don't think this example is right.

Try something like this.

Let the countable family be $\displaystyle A_{m,n}=\left(\frac{2}{m+n},\frac{3}{m}\right)$, $\displaystyle m,n$ naturals.

Then $\displaystyle \limsup A_{m,n} = \max A_{m,n}= A_{1,1}=(1,3)$ but $\displaystyle \liminf A_{m,n} =\lim_{n\rightarrow\infty,m\rightarrow\infty}A_{m, n}=\{0\}.$

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.

$\displaystyle \liminf_{n\to\infty} A_n = \bigcup_{N=1}^\infty \bigcap_{n\ge N} A_n$ and $\displaystyle \limsup_{n\to\infty} A_n = \bigcap_{N=1}^\infty \bigcup_{n\ge N} A_n$

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 $\displaystyle x$ is in $\displaystyle \liminf_{n\to \infty}A_n$ if there exists $\displaystyle N \in \mathbb{N}$ such that for all $\displaystyle n\ge N$, $\displaystyle x \in A_n$. That same point $\displaystyle x$ will be in $\displaystyle \limsup_{n\to \infty}A_n$ if for all $\displaystyle N\in \mathbb{N}$, there exists $\displaystyle n\ge N$ such that $\displaystyle x \in A_n$.

In the OP example, if $\displaystyle B_m = [x,y-e/m]$ and $\displaystyle C_m = [y+e/m,z]$, then $\displaystyle \liminf_{n\to \infty} A_n = \emptyset, \limsup_{n\to \infty}A_n = [x,z]\setminus {y}$, so that could be what you are looking for.