Convergence of a sequence of sets $A_n:=\{1+ \frac{m^2}{n^2}: m \in \mathbb{N} \}$

Question

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.

Answer 1

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.