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?