$f_i$ are smooth functions in $\mathbb{R}^n$ with $\operatorname{supp} f_i$ compact. Show that there are positive real numbers $\{ \lambda_i \}$ s.t. $\sum_i \lambda_i f_i$ is a smooth function.
I have tried to consider the partitions of unity when I see $\{\mathbb{R}^n-\operatorname{supp}f_i\}$ as an open covering of $\mathbb{R}^n$, but I am stuck. What I want to know most is how to find these positive real numbers $\{ \lambda_i \}$? Do they have any special meaning?
Thanks in advance.