Thanks!
I was going through this proof, and it mostly made sense to me. However, I don't understand where in the proof, boundedness of the open set $U$ is required. The construction of a smooth partition of unity does not require a bounded open cover, and the locally finite cover constructed by Evans can also be applied to an unbounded open set $U$. Can someone explain where the proof fails?
The original proof by Meyers and Serrin does not need boundedness.
https://www.semanticscholar.org/paper/H-%3D-W.-Meyers-Serrin/e7a5f41b547afe5b8f38860436446a123c495738?utm_source=direct_link