Taking a bizarre limit Consider the set of integers, $\Bbb{Z}$. Now consider the sequence of sets which we get as we divide each of the integers by $2, 3, 4, \ldots$.
Obviously, as we increase the divisor, the elements of the resulting sets will get closer and closer.
Question: In the limit as $\text{divisor}\to\infty$, what will the "limiting" set be? 
(I don't think it could be $\Bbb{R}$.)

Let $A_n=\{x/n:x\in\mathbb{Z}\}$ with $n$ any integer greater than $1$ then it is easy to see that $\mathbb{Z}\subset A_n\subset \mathbb{Q}$.  We claim that
$$\limsup_n A_n= \bigcap_{N=1}^\infty \left( \bigcup_{n\ge N} A_n \right)=\mathbb{Q}$$ and $$\liminf_n A_n = \bigcup_{N=1}^\infty \left(\bigcap_{n \ge N} A_n\right)=\mathbb{Z}.$$
See set-theoretic limit for more details about  the limit of a sequence of sets.
