 Local compactness plus uniform tail control

Suppose $u_j\to u$ strongly in $L^2(B_R)$ for every $R>0$, and assume uniform tail controlThen $u\in L^2({\mathbb{R}}^n)$ and $u_j\to u$ strongly in L2 space on the whole space. Local convergence and the tail bound first bound the global L2 norms; Fatou's lemma gives the same tail bound for $u$. Splitting the error givesChoose $R$ to make the last two terms uniformly small, then increase $j$. Local Rellich-Kondrachov compactness theorem arguments and a diagonal subsequence supply the first hypothesis in many Sobolev space problems; the tail condition rules out loss of compactness at infinity.