Thread: Union of infinite number of sets is not empty

1. Union of infinite number of sets is not empty

Let [], be closed intervals with [] [] for all . Prove that

I know it has something to do with compactness, but I am not sure how to proceed.

2. Originally Posted by Pythagorean12
Let [], be closed intervals with [] [] for all . Prove that

I know it has something to do with compactness, but I am not sure how to proceed.
What have you tried?!

3. The requirement that $\displaystyle \left( {\forall m,n} \right)\left[ {[a_n ,b_n ] \cap [a_m ,b_m ] \ne \emptyset } \right]$ implies that $\displaystyle \left( {\forall m,n} \right)\left[ {a_n \leqslant b_m } \right]$.

So $\displaystyle A = \left\{ {a_n :n \in \mathbb{Z}^ + } \right\}$ has a least upper bound. Call it $\displaystyle \alpha$.

Show that $\displaystyle \alpha \in \bigcap\limits_n {\left[ {a_n ,b_n } \right]}$.

4. The statement implies that The bounded colsed interval collection has finite intersection property,thus the intersection is nonempty.