Basic topology proof of closed interval in R

[lurifax1]

Let $$\{ [a_j, b_j]\}_{j\in J}$$ be a set of (possibly infinitely many closed intervals in R whose intersection cannot be expressed as a disjoint union of subsets of R. Prove that $$\bigcup\limits_{j \in J} {\{ [{a_j},{b_j}]\} }$$ is a closed interval in R.

I don't understand how to attack this, and would appreciate an example or a push in the right direction! This is the first proof exercise I'd had in this course and I'm pretty lost.

 

[Dick]

I think your problem statement is incomplete. Every subset of R can be expressed as a disjoint union of sets. I think the problem needs to say disjoint union of some particular kind of sets. What kind?