Previous Question; Does upper limit and lower limit exist for any sequence in $\mathbb{R}$?
For every sequence $\{s_n\}$ in $\mathbb{R}$, $\{x\in \overline{\mathbb{R}}| s_{n_k} →x\}$ is nonempty.
However, how do i prove the existence of $\sup \{x\in \overline{\mathbb{R}}| s_{n_k} →x\}$ and $\inf \{x\in \overline{\mathbb{R}}| s_{n_k} →x\}$?
Example; $a_{2n}=-n^2$ and $a_{2n+1}=(1/2)^n$
$b_{3n}=-n^2$ and $b_{3n+1}=(1/2)^n$ and $b_{3n+2}=n^2$.
What kind of mathematical property do they share? I don't know where to start my argument.