Example: Can you find infinitely many closed sets such that their intersection is empty and such that each set is contained in its predecessor ? That is, can you find sets A j such that A j+1 A j and A j= 0 ?
It is easy to simply find some closed sets with empty intersection. For example, the intersection of all intervals of the form [n, n+1] is certainly empty.

To find sets contained in one another is slightly more complicated. We might try sets of the form A j = [0, 1 / j] for all j. Then A j+1 A j, but their intersection contains the point {1}.

Let A j = [j, ). Then A j+1 A j, because [j + 1, ) [j, ). But by the Archimedian principle, the intersection of all sets is empty.

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