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 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?