Question 9
Let $a_{0}$ and $a_{n}$ be different divisors of the natural number $m\gt 1$, and let the sequence of natural numbers $a_{0}, a_{1}, a_{2}, \ldots, a_{n}$ satisfy
$$
a_{i+1}=\left|a_{i} \pm a_{i-1}\right| \quad \text{ for } 0\lt i\lt n
$$
If NZD is $\left(a_{0}, \ldots, a_{n}\right)=1$, prove that there is a term in the sequence that is less than $\sqrt{m}$.
$$
a_{i+1}=\left|a_{i} \pm a_{i-1}\right| \quad \text{ for } 0\lt i\lt n
$$
If NZD is $\left(a_{0}, \ldots, a_{n}\right)=1$, prove that there is a term in the sequence that is less than $\sqrt{m}$.