Let $U= \{ (-n,n): n=1,2,3 . . . \}$ be an open cover for $\mathbb{R}$. I need to construct a smooth partitions of unity subordinate to the open cover $U$ .
Now, I know from the definition that I need to find a collection of functions $\{\phi_{\alpha} :M \rightarrow \mathbb{R}\}_{\alpha \in A}$ ($M$ is a smooth manifold) which satisfies the following properties -
1) $0 \leq \phi_{\alpha}(x) \leq 1 $ for every $\alpha \in A , x \in M$
2) support $\phi_{\alpha} \subset U_{\alpha}$
3) $\{ $Support$ \space \phi_{\alpha}\}$ is locally finite
4) $\sum_{\alpha \in A} \phi_{\alpha}(x) = 1$ for every $x \in M$
But I am unable to think of any such kind of set of functions. Any kind of help is appreciated!