# A simple proof of non-convergence of a sequence

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.