Question 176
Let $f: \mathbb{Z} \to \mathbb{R}$ be a function such that
$$
f(n+2) = (f(n+1) + f(n)) / 2
$$
for every integer $n$. Suppose that $f$ is bounded, that is, there exists a real number $F$ such that $-F \leqslant f(n) \leqslant F$ for all $n$.
Prove that $f$ is a constant function.
$$
f(n+2) = (f(n+1) + f(n)) / 2
$$
for every integer $n$. Suppose that $f$ is bounded, that is, there exists a real number $F$ such that $-F \leqslant f(n) \leqslant F$ for all $n$.
Prove that $f$ is a constant function.