I've seen this problem before, but can't remember how to finish it:
Define an indexed family of sets $ \{A_i : i \in \mathbb{N} \}$ in which for any $m,n\in \mathbb{N}, A_m \cap A_n \not= \emptyset$ and $\bigcap A_i = \emptyset$. The closest I came was something to the effect of $A_i = \{(0, 1/n): n \in \mathbb{N} \}$, but I know that doesn't meet the last criterion.
Suggestions on how to fix/finish it? Am I even as close as I think I am?