Let $I_\delta:=(-\delta,\delta)$ where $\delta>0$. Let $\{v_\epsilon(x)\}_{\epsilon>0} \subset W^{1,2}(-1,1)$ be a sequence such that $0\leq v_\epsilon\leq 1$ and $v_\epsilon(x)\to1$ a.e. (you may just think $v_\epsilon$ is continuous.)
It is already known that $$ \limsup_{\delta\to 0}\,\left[\limsup_{\epsilon\to 0}\left(\inf_{x\in I_\delta}\left\{v_\epsilon(x)+\inf_{y\in(-\delta,x)}(1-v(y))+\inf_{y\in(x,\delta)}(1-v(y))\right\}\right)\right]=0\tag 1 $$
I wish to show the following from $(1)$: (the sequence could be countable or uncountable)
There exists (sub)sequences $\{a_\epsilon\}$, $\{b_\epsilon\}$, and $\{c_\epsilon\}$ such that
- $a_\epsilon<b_\epsilon<c_\epsilon$ and $\lim_{\epsilon\to 0}a_\epsilon = \lim_{\epsilon\to 0}b_\epsilon = \lim_{\epsilon\to 0}c_\epsilon=0$.
- $\lim_{\epsilon\to 0} v_\epsilon(a_\epsilon)=\lim_{\epsilon\to 0}v_\epsilon(c_\epsilon)=1$ and $\lim_{\epsilon\to 0}v_\epsilon(b_\epsilon)=0$.
For a moment I think it might be the direct consequence from $(1)$. But I can't come up with a nice proof... any help is really welcome!