Let $\varphi$ be infinitely divisible, and for every $n \in \mathbb{N}$, let $\varphi_{n}$ be a CFP with $\varphi_{n}^{n}=\varphi$. Use Lévy's continuity theorem to show that $\varphi_{n} \stackrel{n \rightarrow \infty}{\longrightarrow} 1$ uniformly on compact sets $\varphi_{n} \stackrel{n \rightarrow \infty}{\longrightarrow} 1$. Conclude that $\varphi(t) \neq 0$ for all $t \in \mathbb{R}$.