Interactive Real Analysis
- part of Examples 5.2.7(b):
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 ?
Back
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.
A j and
A j= 0 ?
Back
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 {0}.
Let
A j = [j, 
).
Then
A j+1
A j,
because [j + 1, )
[j, ).
But by the Archimedian principle, the intersection of all
sets Aj is empty.