Let $\{s_n\}$ be a sequence of reals. Let $E=\{x\in \overline{\mathbb{R}}|s_{n_k}→x\}$
($\overline{\mathbb{R}}$ denotes extended real number)
Definition of upper limit of $\{s_n\}$ is $\sup E$.
I know that if $E$ is nonempty, it is well-defined.
If $\{s_n\}$ is bounded, then $E$ is nonempty.
However, how do i show that $E$ is nonempty when $\{s_n\}$ is not bounded?
If $\{s_n\}$ is not bounded, $\forall M>0$, there exists $N\in \mathbb{N}$ such that $|s_N|>M$.
It's obvious that at least one of 'set of $s_n$ such that $s_n > M$' and 'set of $s_n$ such that $s_n < M$' must be infinite, but how do i show this?