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$w_n$s are a sequence of functions such that $||w_n||_{L^2(\mathbb{R}^N)}=1$. Thus there exists a subsequence which weakly converges to $w$, say. Further $w_n$ satisfies that $\forall\epsilon>0$, $\exists y_n(\epsilon)$ such that $\int_{y_n+B_R}|w_n|^2\geq 1-\epsilon$ for some $R>0$. Define $\tilde{w_n}(.)=w_n(.+y_n)$. Now I can see that for a subsequence $\tilde{w_n}\rightarrow \tilde{w}$ strongly in $L^2(B_R)$. How do we see the same convergence in $L^2(\mathbb{R}^N)$?.

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