2022 IMO Shortlist A1

Let $\left(a_{n}\right)_{n \geq 1}$ be a sequence of positive real numbers with the property that $$\left(a_{n+1}\right)^{2}+a_{n} a_{n+2} \leq a_{n}+a_{n+2}$$ for all positive integers $n$. Show that $a_{2022} \leq 1$. (Nigeria)

设 $\left(a_{n}\right)_{n \geq 1}$ 为正实数序列，其属性为 $$\left(a_{n+1}\right)^{2}+a_{n} a_{n+2} \leq a_{n}+a_{n+2}$$ 对于所有正整数 $n$。显示 $a_{2022} \leq 1$。 (尼日利亚)