Intersection of a family of non-empty closed intervals

I've been trying to make sense of this in my head. I know that the nested interval theorem proves the following if the intervals are bounded, but what if they are unbounded? I apologize for my ignorance to type mathematical symbols on this editor.

Let I be a family of non-empty closed intervals, and suppose that I has the finite intersection property. Is it true that the intersection of all I is necessarily non empty?

Solution 1:

No. Consider the family of unbounded intervals defined by $$I_n = [n,+\infty),\qquad n=1,2,\dots$$ On the one hand, the intersection of a finite number of these intervals is nonempty. Explicitly, the intersection of the intervals $I_{n_1},I_{n_2},\dots,I_{n_k}$ is $$I_{n_1}\cap I_{n_2}\cap \cdots \cap I_{n_k} = I_{\max(n_1,n_2,\dots,n_k)}\neq\varnothing.$$ On the other hand, the intersection of all the intervals is empty because any number is contained in only finitely many of them.