If you had a finite subset of the collection, then the smallest left-hand endpoint would a number .
But and .
Therefore, the subcollection does not cover .
The statement that set is compact means every open covering has a finite subcover.
Having one open cover which has a finite subcover does not imply that the set is compact.
Consider:
That is clearly an open covering of and the collection contains a finite subcover.
But is still not compact.