Here's an outline: Each $a_j$ is of the form $\exp(\pi i b_j)$. Each $b_j$ can be approximated by a rational number. Then high powers of $a_j$ can be seen to get arbitrarily close to $1$ cyclically, using least common denominators of the $b_j$. This makes the total close to $m$.
And if $k$ of the $a_j$ are not equal to $1$, then high powers of those $a_j$ will cyclically have negative real part, making the total have real part $\le m-k$. Since the limit exists, the only resolution to this is that $k=0$ and they are all equal to $1$.
Again, this is just an outline. All the bits about cycling and closeness would need to be formalized for a solid argument.