Question 9

Let $${x_n = - \left(\frac{n+2}{n + 1}\right)}$$, if $n$ is odd and $${x_n = \frac{1}{\sqrt{n}}}$$ if $n$ is even.

Let $\mathbb{X} = \{ x_n: n \in \mathbb{N}\}$.
For any $k \in \mathbb{N}$, put $\mathbb{X}_k = \{ x_n: n \ge k, n \in \mathbb{N}\}$.
Let $a_k = \inf(\mathbb{X}_k)$ and $b_k = \sup(\mathbb{X}_k)$, for any $k \in \mathbb{N}$.
Put $\mathbb{J}_k = [a_k, b_k]$, for any $k \in \mathbb{N}$.
Prove that $\{ \mathbb{J}_k: k \in \mathbb{N}\}$ is a nested sequence of intervals.
Do there exist three subsequences of $\mathbb{X}$ with different limits?
Explain your answer.