Question 165
A sequence $\left(a_{n}\right)_{n \geqslant 0}$ is defined as follows: we choose $a_{0}, a_{1} \in \mathbb{N}^{*}$, and for every integer $n \geqslant 0$, we set
$$
a_{n+2}=\left\lfloor\frac{2 a_{n}}{a_{n+1}}\right\rfloor+\left\lfloor\frac{2 a_{n+1}}{a_{n}}\right\rfloor,
$$
where $\lfloor x\rfloor$ denotes the largest integer not exceeding $x$. Prove that there exists $m \in \mathbb{N}$ such that $a_{m}=4$ and $a_{m+1} \in\{3,4\}$.
$$
a_{n+2}=\left\lfloor\frac{2 a_{n}}{a_{n+1}}\right\rfloor+\left\lfloor\frac{2 a_{n+1}}{a_{n}}\right\rfloor,
$$
where $\lfloor x\rfloor$ denotes the largest integer not exceeding $x$. Prove that there exists $m \in \mathbb{N}$ such that $a_{m}=4$ and $a_{m+1} \in\{3,4\}$.