Convergence of a sequence of sets $A_n:=\{1+ \frac{m^2}{n^2}: m \in \mathbb{N} \}$ Let $n \in \mathbb{N}$ and let $A_n$ be the set given by
$$A_n:=\left\lbrace 1+ \frac{m^2}{n^2}: m \in \mathbb{N} \right\rbrace$$
Can you help me to determine if 
$\lim_{n \to \infty} A_n $
exists?
Thank you for any help.
 A: To find whether the limit of a sequence of the sets exists we need to calculate $$ \liminf_{n\to\infty} A_n = \bigcup_{n\ge 1} \bigcap_{k\ge n} A_k$$ and $$ \limsup_{n\to\infty} A_n = \bigcap_{n\ge 1} \bigcup_{k\ge n} A_k$$
If both of these sets are equal then they define $\lim_{n\to\infty} A_n$.
We have, for any $n\in \mathbb N$: \begin{align} \bigcap_{k\ge n}A_k &= \bigcap_{k\ge n} \left\{z: \exists m\in\mathbb N: z=1+\frac {m^2}{k^2} \right\}=\\&=\left\{z: \forall k\ge n \exists m\in\mathbb N: z=1+\frac {m^2}{k^2} \right\}= \\ &= \left\{1+q^2: \forall k\ge n \exists m\in\mathbb N: q=\frac mk \right\} =\\&= \left\{1+q^2: \forall k\ge n : kq \in\mathbb N\right\} =\\&= \left\{1+q^2: q \in\mathbb N\right\}\end{align}
so
$$ \liminf_{n\to\infty} A_n = \bigcup_{n\ge 1} \bigcap_{k\ge n} A_k = \left\{1+q^2: q \in\mathbb N\right\}$$
We also have
$$ \forall n\in\mathbb  N \quad \forall q\in\mathbb Q_+ \quad \exists k\ge n \quad\exists m\in\mathbb N : q=\frac mk$$
$$ \forall n\in\mathbb  N \quad \forall q\in\mathbb Q_+ \quad \exists k\ge n : 1+q^2\in A_k$$
$$ \forall n\in\mathbb  N \quad \forall q\in\mathbb Q_+ :  1+q^2\in \bigcup_{k\ge n} A_k$$$$ \forall n\in\mathbb N :\{1+q^2:q\in\mathbb Q_+\} \subset \bigcup_{k\ge n}A_k$$
On the other hand it is obvious that
$$ \forall k\in\mathbb N: A_k \subset \{1+q^2:q\in\mathbb Q_+\}$$
so
$$ \forall n\in\mathbb N: \bigcup_{k\ge n}A_k \subset \{1+q^2:q\in\mathbb Q_+\}$$
and that means that
$$ \forall n\in\mathbb N: \bigcup_{k\ge n}A_k = \{1+q^2:q\in\mathbb Q_+\}$$
$$ \bigcap_{n\ge 1} \bigcup_{k\ge n}A_k = \{1+q^2:q\in\mathbb Q_+\}$$
$$ \limsup_{n\to\infty} A_n = \{1+q^2:q\in\mathbb Q_+\}$$
We have 
$$ \liminf_{n\to\infty} A_n = \left\{1+q^2: q \in\mathbb N\right\} \neq \left\{1+q^2:q\in\mathbb Q_+\right\} = \limsup_{n\to\infty} A_n $$
so $\lim_{n\to\infty} A_n $ does not exist.
