Terminology: Upper limit and lower limit. Let $(x_n)$ be a sequence of real numbers. It defines:
$\limsup\limits_{n \rightarrow \infty} x_n=\lim\limits_{n \rightarrow \infty} (\sup_{m≥n} x_m)=\inf_{n≥1}(\sup_{m≥n}x_m)$
How should I understand the notation of the definition? Specifically the $m≥n$ and $n≥1$ terms.

$\inf_{\text{condition}}(\text{quantity})$ (and similar for $\sup$) needs to be understood as $\inf\{\text{quanity}\mid\text{condition}\}$, i.e as the infimum of the set of the quantities satisfying the condition (and similar for the supremum).
Thus, $\sup_{m\ge n}x_m$ is actually $\sup\{x_m\mid m\ge n\}=\sup\{x_n,x_{n+1},x_{n+2},\ldots\}$. Similarly:
$$\begin{array}{rcl}\inf_{n\ge 1}\sup_{m\ge n}x_m&=&\inf\{\sup\{x_m\mid m\ge n\}\mid n\ge 1\}\\&=&\inf\{\sup\{x_1,x_2,x_3,x_4\ldots\},\sup\{x_2,x_3,x_4,\ldots\},\sup\{x_3,x_4,\ldots\},\ldots\}\end{array}$$
Note as a set gets smaller, it's supremum gets smaller and its infimum gets bigger.
