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 $\Bbb{R}$.)

The typical way to define limits of sets is via
$$\liminf_{n\to\infty} A_n = \bigcup_{n\geq 1} \bigcap_{k \geq n} A_k \\ \limsup_{n\to\infty} A_n = \bigcap_{n\geq 1} \bigcup_{k\geq n} A_k$$
Using these and $A_n = f_n(\mathbb{Z})$ where $f_n(x) = x/n,$ we have
$$\liminf_{n\to\infty} A_n = \mathbb{Z} \\ \limsup_{n\to\infty} A_n = \mathbb{Q} $$
In particular, the limit doesn't exist.
