Question 2
Show that $r=2$ is the largest real number $r$ with the following property: if a sequence $a_1,a_2,\ldots$ of positive integers satisfies
$$a_n\le a_{n+2}\le\sqrt{a_n^2+r a_{n+1}}$$
for every positive integer $n$, then there is a positive integer $M$ such that $a_{n+2}=a_n$ for every $n\ge M$.
$$a_n\le a_{n+2}\le\sqrt{a_n^2+r a_{n+1}}$$
for every positive integer $n$, then there is a positive integer $M$ such that $a_{n+2}=a_n$ for every $n\ge M$.