Let $\left\{ x_{n}\right\} _{n}$ be a sequence. Define
$E_0=${$r\in\mathbb{R}$:$\lim_{k\rightarrow\infty}$ $x_{n_{k}}=r$}.
If $\left\{ x_{n}\right\} _{n}$ has a subsequence converging to $\pm \infty$ we add this to $E_0$ to obtain $E$. So,
$E=${$x\in\mathbb{R}\cup${$\pm \infty$}:$\lim _{k\rightarrow \infty }x_{n_{k}}=x$}.
Remark. If $\left\{ x_{n_{k}}\right\} _{k}$ is increasing then sup$_{k} $$x_{n_{k}}\in E$, if not inf$_{k}$ $x_{n_{k}}\in E$.
I couldn't understand this remark that can you explain more clearly?
So $\{x_{n_k}\}_k$ is increasing. If this series is bounded, we know it will converge to a number $M$, otherwise, it will go to infinity. Either way, we could say $\sup_k\{x_{n_k}\}=\lim_{k\rightarrow \infty}=x,\,x\in \mathbb R \cup\{-\infty,+\infty\}$.
Similar logic could be applied to decreasing and $\inf$.
However, I do not think the "if not" in the remark would be correct - not increasing does not mean decreasing. If it is oscillating, $\inf$ always exists in $\mathbb R \cup\{-\infty,+\infty\}$, but $\lim$ does not necessarily exist.