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