I’m looking for a proof of the following fact:

Suppose $a_n$ is a sequence of complex numbers of the form

$$a_n = sum_{j=1}^m c_j alpha_j^n$$

where $m$ is a positive integer and the $c_j, alpha_j$ are complex numbers with $|alpha_j|=1$ for each $j=1, ldots, m$. Assume also that the $alpha_j$ are distinct. Then, $a_n$ can not converge to $0$.

I think I can cobble together a proof using dynamics on the $m$ torus (i.e. using irrational rotations) but was hoping to find an easier proof. Any suggestions would be greatly appreciated – thanks.