I have the following question: is it true, whatever definition of a limit superior and a limit inferior of a sequence (say, in $\mathbb{R}$) we take, we would have
$\lim\sup x_n = \infty$ if $(x_n)$ is not bounded above,
$\lim\inf x_n = -\infty$ if $(x_n)$ is not bounded below?
I think this is true, but want to check if I'm correct. My reasoning:
Suppose that $(x_n)$ is not bounded above (resp., below). Then $\sup_{k \geq n} x_k = \infty$ (resp., $\inf_{k \geq n} x_k = - \infty$) for all $n \in \mathbb{N}$, because, were we to assume that, for any $n \in \mathbb{N}$, $\sup_{k \geq n} x_k < \infty$ (resp., $\sup_{k \geq n} x_k > - \infty$), we would have $\infty = \sup\{ x_n \mid n \in \mathbb{N} \} = \max\{x_0, ..., x_{n-1}, \sup_{k \geq n} x_n \} \in \mathbb{R}$ (resp., $-\infty = \inf\{x_n \mid n \in \mathbb{N}\} = \min\{x_0,...,x_{n-1},\inf_{k\geq n} x_n\} \in \mathbb{R}$), which is an obvious contradiction.
Now it should be clear that a limit of a constant sequence should be the constant in question, no matter what the definition of convergence we take in the extended number line $\overline{\mathbb{R}}$.