lim sup and sup inequality Is it true that for a sequence of functions $f_n$
$$\limsup_{n \rightarrow \infty }f_n \leq \sup_{n} f_n$$
I tried to search for this result, but I couldn't find, so maybe my understanding is wrong and this does not hold.

The inequality
$$\limsup_{n\to\infty}a_n\leq\sup_{n\in\mathbb{N}}a_n$$
holds for any real numbers $a_n$, because the definition of $\limsup$ is
$$\limsup_{n\to\infty}a_n:=\lim_{m\to\infty}\left(\sup_{n\geq m}a_n\right)$$
and for any $n\in\mathbb{N}$, we have
$$\left(\sup_{n\geq m}a_n\right)\leq\sup_{m\in\mathbb{N}}a_n$$
(if the numbers $a_1,\ldots,a_{m-1}$ are less than or equal to the supremum of the others, both sides are equal, and if not, then the right side is larger).Therefore
$$\limsup_{n\to\infty}f_n(x)\leq \sup_{n\in\mathbb{N}}f_n(x)$$
holds for any real number $x$, which is precisely what is meant by the statement
$$\limsup_{n\to\infty}f_n\leq \sup_{n\in\mathbb{N}}f_n.$$
