$\lim \inf, \lim \sup$ of a sequence $A_n=\{{\frac{m}{n}} | m\in\Bbb{N} \}$

Question

Let $A_n=\{{\frac{m}{n}} | m\in\Bbb{N} \}$, $n\in \Bbb{N}$, where $\Bbb{N}$ are non-negative integers. How do I find $\liminf_{n\to\infty} A_n$ and $\limsup_{n\to\infty} A_n$. Developing the sequence, I would say that $\limsup$ is equal to $\Bbb{N}$. Although I can't find a formal way to develop it.

Answer 1

Yes, $\liminf_nA_n=\mathbb N$. That's so, because if $k\in\mathbb N$, then $k$ belongs to every $A_n$. On the other hand, each $A_n$ is a set of positive rational numbers. But if $q\in\mathbb Q\setminus\mathbb N$, then $q=\frac ab$ with $\gcd(a,b)=1$ and $b>1$. But then, if $\gcd(b,n)=1$, $q\notin A_n$. So, it is not true that $q\in A_n$ if $n\gg1$. In other words, $q\notin\liminf_nA_n$.

And $\limsup_nA_n=\{q\in\mathbb Q\mid q>0\}$ (every positive rational belongs to infinitely many $A_n$'s).