## Proposition 5.2.8: Intersection of Nested Compact Sets

Suppose

*{ A*is a collection of sets such that each_{j}}*A*is non-empty, compact, and_{k}*A*. Then_{j+1}A_{j}*is not empty.***A**= A_{j}### Proof:

Each*A*is compact, hence closed and bounded. Therefore,

_{j}**A**is closed and bounded as well, and hence

**A**is compact. Pick an

*a*for each

_{j}A_{j}*j*.

Then the sequence
*{ a _{j} }*
is contained in

*A*. Since that set is compact, there exists a convergent subsequence

_{1}*{ a*with limit in

_{jk}}*A*.

_{1}
But that subsequence, except the
first number, is also contained in
*A _{2}*.
Since

*A*is compact, the limit must be contained in

_{2}*A*.

_{2}
Continuing in this fashion, we see
that the limit must be contained in every
*A _{j}*, and hence it is also contained in their
intersection

**A**. But then

**A**can not be empty.