Let
![r](/media/m/3/d/f/3df7cc5bbfb7b3948b16db0d40571068.png)
be a positive integer, and let
![a_0, a_1, \ldots](/media/m/f/8/8/f889e363523f30ec1df9db82353b06ed.png)
be an infinite sequence of real numbers. Assume that for all nonnegative integers
![m](/media/m/1/3/6/1361d4850444c055a8a322281f279b39.png)
and
![s](/media/m/9/0/8/908014cbadb69e42261a56b450a375b9.png)
there exists a positive integer
![n \in [m + 1, m + r]](/media/m/5/d/7/5d7b59438e77c03c392bfe9a83659e20.png)
such that
![a_m + a_{m+1} + \cdots + a_{m+s} = a_n + a_{n+1} + \cdots + a_{n+s} \text{.}](/media/m/e/5/c/e5cca3123302f9960d144178ae18fb9f.png)
Prove that the sequence is periodic, i.e. there exists some
![p \geq 1](/media/m/f/c/8/fc8a815b8dd2ad8a7bd855ecf4183563.png)
such that
![a_{n+p} = a_n](/media/m/e/9/d/e9d669e095deed1b0bbe62e05988ceea.png)
for all
![n \geq 0](/media/m/0/5/2/052ae3f202c82fff970ac34992a6c9d3.png)
.
%V0
Let $r$ be a positive integer, and let $a_0, a_1, \ldots$ be an infinite sequence of real numbers. Assume that for all nonnegative integers $m$ and $s$ there exists a positive integer $n \in [m + 1, m + r]$ such that $$
a_m + a_{m+1} + \cdots + a_{m+s} = a_n + a_{n+1} + \cdots + a_{n+s} \text{.}
$$
Prove that the sequence is periodic, i.e. there exists some $p \geq 1$ such that $a_{n+p} = a_n$ for all $n \geq 0$.