Question 3
The sequence $c_{0}, c_{1}, \ldots, c_{n}, \ldots$ is defined by $c_{0}=1$, $c_{1}=0$ and $c_{n+2}=c_{n+1}+c_{n}$ for $n \geq 0$. Consider the set $S$ of ordered pairs $(x, y)$ for which there is a finite set $J$ of positive integers such that $x=\sum_{j \in J} c_{j}$, $y=\sum_{j \in J} c_{j-1}$. Prove that there exist real numbers $\alpha$, $\beta$ and $m$, $M$ with the following property: An ordered pair of nonnegative integers $(x, y)$ satisfies the inequality
$$
m\lt \alpha x+\beta y\lt M
$$
if and only if $(x, y) \in S$.
N. B. A sum over the elements of the empty set is assumed to be $0$.
$$
m\lt \alpha x+\beta y\lt M
$$
if and only if $(x, y) \in S$.
N. B. A sum over the elements of the empty set is assumed to be $0$.