I want to prove the following statement. Let $$\mu=\sum\limits_{\lambda\in\Lambda}\mu(\lambda)\delta_{\lambda},\mu(\lambda)\neq0$$ a measure in $\mathbb{R}^{n}$ with $\Lambda$ a uniform discrete set. Then $\mu$ is a temperate distribution if and only if $$|\mu(\lambda)|\leq C(1+|\lambda|^{N}),\lambda\in\Lambda$$ for some positive Constants $C$ and $N$.
I managed to solve the $\leftarrow$ direction. For the other direction my thought was something to try with $|\mu(\lambda)|>C(1+|\lambda|^{N})$ . Because then $|\mu(\lambda)|$ would be grow faster then a polynomial and then the sum would be infinite but I'm not sure how to write that down.