TITLE: question about limit and series
QUESTION [6 upvotes]: consider  following hypotheses 


$ m\in\mathbb N$
$c\in \mathbb C\,$ ,$\, \; a_j\in \mathbb C$
$a_j\in \mathbb C\;$ , $\;|a_j|=1,\;\forall\;1\le j\le m$


if $$\lim\limits_{n\to+\infty}\sum_{j=1}^{m}a^n_j=c$$
then how to prove  $ c=m$ and $\forall1\le j\le m\;$ , $\;a_j=1$ .
Thanks in advance

REPLY [3 votes]: 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.