Let $\{u_{i}\}$ be sequence of smooth function defined on $\Bbb{R}^n$ such that $\|u_i\|_{L^2(\Bbb{R}^n)}$ is uniformly bounded in $i$ and $\|\nabla u_{i} \|_{L^{^2}(\Bbb{R}^n)}$ is also uniformly bounded. And $\lim _{R\to \infty}\int_{\Bbb{R}^n \setminus \{B_R\}}|u_i|^2 = 0$ uniformly in $i$
How Can I use Arzela Ascoli theorem to extract the convergence subsequence?
I know one version for $L^p$ is if $1 \le p<\infty$, and
(1) $$ {\displaystyle \lim _{r\to \infty }\int _{|x|>r}\left|u_i\right|^{p}=0} \ \text{ uniformly in }i$$
(2) if $$\lim_{|h|\to 0}\|\tau_h f -f\|_{p} = 0 \ \text{ uniformly in }i$$ then there exist a convergent subsequence in $L^p$, however I have no idea how to apply this theroem.
Therefore I do as follows:
First $$u_i(x+h) - u_i(x) = \int_{0}^1 \frac{d}{dt}u_i(x+th) dt $$
therefore :
$$|u_i(x+h) - u_i(x)|^2 \le \int_0^1 |\nabla u_i(x+th) ||h|dt$$
Hence :
$$\|\tau_{h} u_i - u_i\|_{L^2}^2 \le|h|^2 \int_{\Bbb{R}^n} \int_{0}^1 |\nabla u_i (x+th )| = |h|^2 \int_{0}^1 \int_{\Bbb{R}^n} |\nabla u_i (x+th )| = |h|^2\|\nabla u_i\|_{L^2}^2 $$
therefore translation is Lip continuous in $L^2$,and the sequence is Equitight by assumption, Which is exactly the condition we want,
is my understanding correct?
If $U$ is some bounded smooth domain(instead of $\Bbb{R}^n$), I can do as follows, first uniformly bounded in $H^1$ implies weak convergent subsequence , therefore apply the sobolev embedding $H^1 \to L^2$ there exist strong convergent subsequence in $L^2(U)$ , is my understanding correct?
As shown in daw's comment,Equitight condition is necessary for the statement holds.