Question 15
Given $2021$ distinct positive integers $a_1, a_2, \dots, a_{2021}$. Define the sequence $\{a_n\}$ inductively as follows: for each integer $n \ge 2022$, $a_n$ is the smallest positive integer different from $a_1, a_2, \dots, a_{n-1}$ and not dividing the product $a_{n-1}a_{n-2}\dots a_{n-2021}$. Prove: there exists a positive integer $M$, such that all integers greater than or equal to $M$ appear in $\{a_n\}$.