Let $(M, \mu)$ be a weighted Riemannian manifold with the Heine-Borel-property and $\Omega$ be a open subset of $M$. I am looking for a sequence of functions $f_{n}$ such that $0\leq f_{n}\leq 1$ in $\Omega$, $f_{n}$ has compact support in $\Omega$ for all $n$, $f_{n}\uparrow 1$ in $\Omega$ and $$\int_{\Omega}{|\nabla f_{n}|^{2}d\mu}<\infty.$$ My idea is to take a sequence of smooth cut of functions $\varphi_{k}\in C_{0}^{\infty}(\Omega)$ and set $$f_{n}=\sum_{k=1}^{n}{\varphi_{k}}.$$ Then $f_{n}$ has compact support for all $n$, $0\leq f_{n}\leq 1$, but I only have that $f_{n}\uparrow 1$ in some open neighborhood of some compact set $K\subset \Omega$. Can someone help?
Thanks in advance!
Suggestion/Hint: choose a sequence of compact sets $K_n $ such that $K_n\subset K_{n+1}$ and such that $\cup_n K_n = \Omega$ Then define a sequence $(f_{n,k})_k$ on each $K_n$ as you did in your question and look at $g_n:= f_{n,n}$