Let $H$ be any function space on $\mathbb{R}^N$ . Let $u_n,v_n\in H$ are sequences of functions with compact support such that $\text{dist}(\text{supp}(u_n),\text{supp}(v_n))\to\infty$ as $n\to\infty$ . Define $w_n=u_n+v_n$ . I want to prove that $w_n$ cannot be relatively compact .
I think that we can always find some $\chi_n\in C_c^\infty(\mathbb{R}^N)$ with given norm in $H$ such that $\displaystyle\int_{\mathbb{R}^N}w_n\chi_n=0$ for large $n$ since $w_n$ is also of compact support . But how to prove $w_n$ cannot have a convergent subsequence ? Any help is appreciated .