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.

this is true. take any x, then for all n $f_n (x) \leq \sup_n f_n(x)$ by definition of supremum, hence also every limit point of any subsequence $f_{n'} (x)$ is also smaller or equal to $\sup_n f_n{(x)}$
