It is true that given any sequence of numbers, there is a monotone subsequence within. I thought obtaining a monotone subsequence out of a sequence of functions might be too much to ask. But given a sequence of $L^1$ functions, could we say that it admits a monotone subsequence?
EDIT: My motive for asking this was to decide the following problem. Given a sequence of non-negative measurable functions $\{f_n\}$, it is known that for all measurable sets $E$, $\int_Ef_n\to 0$. Could we conclude from here that $f_n\to 0$?