The sequence
![c_{0}, c_{1}, . . . , c_{n}, . . .](/media/m/7/8/a/78a4f21bbf21432f8b99c2fa49199f97.png)
is defined by
![c_{0}= 1, c_{1}= 0](/media/m/b/6/e/b6e1bddb6cfe1b9a220722e30bd938cb.png)
, and
![c_{n+2}= c_{n+1}+c_{n}](/media/m/5/3/d/53da2d84979eb635ecc3fd3af0b21df4.png)
for
![n \geq 0](/media/m/0/5/2/052ae3f202c82fff970ac34992a6c9d3.png)
. Consider the set
![S](/media/m/c/6/3/c63593c3ec0773fa38c2659e08119a75.png)
of ordered pairs
![(x, y)](/media/m/1/5/2/1520b43353795b60686f7df83802e90a.png)
for which there is a finite set
![J](/media/m/9/0/e/90ef5cc2558381e341da5808eb92126f.png)
of positive integers such that
![x=\sum_{j \in J}{c_{j}}](/media/m/e/f/1/ef19dbd148b0d2833ad70a31272055d2.png)
,
![y=\sum_{j \in J}{c_{j-1}}](/media/m/b/1/8/b186324e7b0e48fc6f06cc5d5e754a44.png)
. Prove that there exist real numbers
![\alpha](/media/m/f/c/3/fc35d340e96ae7906bf381cae06e4d59.png)
,
![\beta](/media/m/c/e/f/cef1e3bcf491ef3475085d09fd7d291e.png)
, and
![M](/media/m/f/7/f/f7f312cf6ba459a332de8db3b8f906c4.png)
with the following property: An ordered pair of nonnegative integers
![(x, y)](/media/m/1/5/2/1520b43353795b60686f7df83802e90a.png)
satisfies the inequality
![m < \alpha x+\beta y < M](/media/m/5/d/f/5df1ba3176ba83aee87a759af5c62fba.png)
if and only if
![(x, y) \in S](/media/m/5/c/1/5c13d88851b9043ae5e073a77b9c5a9f.png)
.
Remark: A sum over the elements of the empty set is assumed to be
![0](/media/m/7/b/8/7b8b0b058cf5852d38ded7a42d6292f5.png)
.
%V0
The sequence $c_{0}, c_{1}, . . . , c_{n}, . . .$ 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 $M$ with the following property: An ordered pair of nonnegative integers $(x, y)$ satisfies the inequality $m < \alpha x+\beta y < M$ if and only if $(x, y) \in S$.
Remark: A sum over the elements of the empty set is assumed to be $0$.