Let $f:\mathbb{R} \to \mathbb{R}$ be a smooth function with $|f'(x)| \leq C$ for all $x$ and $f(0)=0$.

Suppose $u_n \to u$ in $H^{\frac 12}(\partial\Omega)$, where $\Omega$ is a bounded domain of class $C^2$. Here, I use the norm $$|u|_{H^{\frac 12}(\partial \Omega)}^2 = |u|^2_{L^2(\partial\Omega)} + \int_{\partial\Omega}\int_{\partial\Omega} \frac{|u(x)-u(y)|^2}{|x-y|^n}.$$

Does it follow that $f(u_n) \to f(u)$ in $H^{\frac 12}(\partial\Omega)$ (at least for a subsequence)?

Note that $f(u) \in H^{\frac 12}$ by the Lipschitz nature of $f$. The only issue is in the convergence of the seminorm, which I can't seem to prove.