I have the following problem:
Let M be a (non-compact) manifold and $\{\theta_i\}$ a partition of unity subordinate to a countable open cover of relatively compact sets. Then the map $f=\sum_i i\theta_i$ is a proper map
So far I only know that using $f$ is continous, I can show $f$ is proper if I can somehow show that $f^{-1}([0,N])$ is bounded, $\forall N\in\mathbb{N}$(This makes sense because M is considered as an embedded manifold in some $\mathbb{R}^p$).
Any help is appreciated, I'm quite stuck. Thanks everyone!
Let $\{V_i\}$ denote the given countable open cover by relatively compact sets to which the partition of unity $\{\theta_i\}$ is subordinate. Then we claim $f^{-1}([0,N]) \subseteq \bigcup_{i=1}^N V_i$ for each $N \in \mathbb{N}.$
To see this, first note that if $x\in M,$ $$f(x) = \sum_{i=1}^\infty i\theta_i(x) \geq \sum_{i=N+1}^\infty i\theta_i(x) \geq \sum_{i=N+1}^\infty (N+1)\theta_i(x) = (N+1)\left(1 - \sum_{i=1}^N \theta_i(x)\right)$$
If $x \in M \setminus \bigcup_{i=1}^NV_i,$ then $\theta_i(x) = 0$ for each $i=1,\ldots,N$ and therefore $$f(x) \geq (N+1)\left(1 - \sum_{i=1}^N\theta_i(x)\right) = N+1.$$
The claim follows. Since further $$f^{-1}([0,N]) \subseteq \bigcup_{i=1}^N \overline{V_i}$$ shows that $f^{-1}([0,N])$ is contained in a compact set, it is bounded.