A $W^{1,2}_{loc}$ function with uniformly bounded integrals on compact subsets $W^{1,2}$?

Question

Let $M$ be a Riemannian manifold, $\Omega\subset M$ is an open subset, let $f\in W^{1,2}_{loc}(\Omega)$ with uniformly bounded integrals on compact subset, i.e. there exists a $C>0$, such that for any compact subset $K\subset \Omega$ $$ \|f\|_{W^{1,2}(K)} \leqslant C $$ Then $f\in W^{1,2}(\Omega)$?

Answer 1

Yes. There is an increasing sequence of compacts $K_n\subset\Omega$ whose union is $\Omega$, then $\int_{\Omega}(|f|^2+|\nabla f|^2)=\lim_{n\to\infty}\int_{K_n}(|f|^2+|\nabla f|^2)\le C$.