ISL 2022 A1

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