Suppose you have a sequence $ f_n \rightarrow f $ (not necessarily increasing), and suppose that for each $n$ there is a sequence of increasing functions $\phi^{(n)}_{m} \rightarrow f_n$. Is it possible that there is an increasing subsequence of $\phi^{(n)}_{m}$ converging to function $f$?
If someone knows a result that guarantees that this is true can you post the book that contains it or the result?
I would like to know if this argument is true to solve an exercise of measure theory. Actually the sequences $\phi^{(n)}_{m}$ are simple functions.
In general it is not possible. Let for instance $f_n(x)=1/n$ and $\phi^{(n)}_m(x)=1/n-1/(m+n)$ for all $x\in\Bbb R$. Then $f_n$ converges to $0$, $\phi^{(n)}_m$ is increasing and converges to $f_n$ as $m\to\infty$, but it is impossible to have a subsequence of $\phi^{(n)}_m$ increasing and converging to $0$, since $\phi^{(n)}_m(x)>0$ for all $n$, $m$ and $x$.