Let $\{f_{n}(x)\}$ be a sequence of uniformly Lipschitz continuous function on $\mathbb{R}$ such that $|f_{n}(x)|\leq 1$ for all $x$ and all $n$. Also, $\|f_{n}\|_{L^{2}(\mathbb R)}\leq 1$ for all $n$.
The question is: if the sequence $|f_{n}|$ is not increasing on $\mathbb R$, is there any way to get a new sequence, in terms of $|f_{n}|$, which will be increasing on $\mathbb R$, or at leasst on a subset of $\mathbb R$? ( For example, we can divide each $f_{n}$ by it is $L^{2}$-norm to get a new sequence of continuous functions which have norms equal to 1, i.e. normalizing).
If it is hard to answer this question with the given information, what else the sequence should satisfy to have such result?
Edit: the sequence of norms $\{\|f_{n}\|_{L^{2}(\mathbb R)}\}$ is increasing.
