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.
 A: The notation $\sup_{m\geq n} x_m$ is just a short way to write $\sup\{x_m: m\geq n\}=\sup\{x_n,x_{n+1},x_{n+2},...\}$, the supremum of the sequence $(x_m)$ starting from the $n$th element.
Similarly, if we let $y_n=\sup_{m\geq n} x_m$ for every $n\in\mathbb{N}$, then $\inf_{n\geq 1} y_n$ is just $\inf\{y_n: n\in\mathbb{N}\}$.
A: $\inf_{\text{condition}}(\text{quantity})$ (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.
A: At the limit the sequence has an upper bound.
The sequence can do things in the "short-term" or when $n$ is "small" that we will ignore.  But eventually $n$ will be big enough, that increasing $n$ will not have any bearing on the the supremum of the tail of the sequence for values bigger than $n.$
$\limsup\limits_{n \rightarrow \infty} x_n=\lim\limits_{n \rightarrow \infty} (\sup_{m≥n} x_m)$
When $n$ is large and we look at the subsequence when $m>n$ that subsequence has an upper bound and that is the limsup of the sequence.
$\limsup\limits_{n \rightarrow \infty} x_n = \inf_{n≥1}(\sup_{m≥n}x_m)$
This says to look at the upper bound of the entire sequence.  The supremum of the sequence to the right of $n$ is less than or equal to the supremum of the sequence including $n.$  When $n$ gets to be large enough, we have found the smallest value of the sequence to the right of $n.$ And this is also the limsup of the sequence.
The sequence will be arbitrarily close to the limsup infinitely many times.
