I'm very stuck on the following exercise in the book "A Comprehensive Introduction to Differential Geometry V.1" by Michael Spivak: Let $M^m$ be a smooth connected non-compact manifold. Show that $M$ is the union of a sequence of open sets $U_n$ with the following properties:
- $U_n \cap U_{n+1}$ is non-empty for all $n$
- For every compact set $C \subset M$ there is $N$ such that $U_n \cap C$ is empty for all $n>N$
- $U_n$ is diffeomorphic to $\mathbb{R}^m$ for all $n$.
Any help would be greatly appreciated. Thank you.