Consider a sequence of functions $\{u_n\}$ in $H^s(\Omega)$ with $u_n \to u$ in $L^2(\Omega)$ and $\int_{\mathbb{R}^N}\frac{u_n(x) - u_n(y)}{|x - y|^{N + 2s}}dy < \infty$ , for every $n \in \mathbb{N}$ and for some $s \in (0,1)$. Here, $\Omega \subset \mathbb{R}^N$ is a bounded and smooth domain. Moreover, $u_n = 0$ in $\mathbb{R}^N \backslash \Omega$. It is possible to show that there exists some integrable function $K$ such that $$\frac{u_n(x) - u_n(y)}{|x - y|^{N + 2s}} \leqslant K(x,y)$$ ?
My intention is to use Lebesgue's dominated convergence theorem to prove that $\int_{\mathbb{R}^N}\frac{u_n(x) - u_n(y)}{|x - y|^{N + 2s}}dy \to \int_{\mathbb{R}^N}\frac{u(x) - u(y)}{|x - y|^{N + 2s}}dy$ a.e on $\Omega$.