Question 155
Suppose a doubly infinite sequence of real numbers
$$
\dots, a_{-2}, a_{-1}, a_0, a_1, a_2, \dots
$$
has the property that
$$
a_{n+3} = \frac{a_n + a_{n+1} + a_{n+2}}{3}, \quad \text{for all integers } n.
$$
Show that if this sequence is bounded (i.e., if there exists a number $R$ such that $|a_n| \le R$ for all $n$), then $a_n$ has the same value for all $n$.
$$
\dots, a_{-2}, a_{-1}, a_0, a_1, a_2, \dots
$$
has the property that
$$
a_{n+3} = \frac{a_n + a_{n+1} + a_{n+2}}{3}, \quad \text{for all integers } n.
$$
Show that if this sequence is bounded (i.e., if there exists a number $R$ such that $|a_n| \le R$ for all $n$), then $a_n$ has the same value for all $n$.