APMO 2020 Problem 2

Show that $r=2$ is the largest real number $r$ which satisfies the following condition:

If a sequence $a_{1}, a_{2}, \ldots$ of positive integers fulfills the inequalities

$$a_{n} \leq a_{n+2} \leq \sqrt{a_{n}^{2}+r a_{n+1}}$$

for every positive integer $n$, then there exists a positive integer $M$ such that $a_{n+2}=a_{n}$ for every $n \geq M$.