In my homework I am asked to show the existence of a proper function $f:M\rightarrow \mathbb{R}$, and there is a hint which suggests to use a partition of unity. (here proper means that the inverse image of a compact set is compact)
I would like some help since i am not even sure where to start. thanks!
My first answer was not correct. Let's try something else. Consider a partition of unity $\left(U_{i},\varphi_{i}\right)$ subordinate to a countable open cover of $M$ by open sets $U_{i}$ with compact closure. Define $\theta\colon M\rightarrow\mathbb{R}$ by $\theta=\sum_{i=1}^{\infty}i\varphi_{i}$. The function $\theta$ is well-defined due to the properties of partitions of unity. We see that, for every integer $j$ : $$ \theta^{-1}\left(\left[-j,j\right]\right)\subset\bigcup_{i=1}^{j}\left\{ x:\varphi_{i}\left(x\right)\ne0\right\} , $$ which has a compact closure. Hence $\theta^{-1}\left(\left[-j,j\right]\right)$ is a compact set.
Another suggestion:
Cover your manifold with a countable collection of open sets $U_1,U_2,\ldots$, such that the closure $\overline{U}_n$ is compact for every $n$ (why is this even possible?). On every $U_n$ take the constant function $f_n\equiv n.$ Then use a partition of unity.
