indexed family of sets, not pairwise disjoint but whole family is disjoint 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?

How about $A_i =\{n\in\mathbb N\mid n>i\}$? But your choice, more properly written $A_i=(0,1/i)$,  is also fine.
