CN-TST 2021 Problem 14

Given distinct positive integer $a_1,a_2,…,a_{2020}$ . For $n \ge 2021$ , $a_n$ is the smallest number different from $a_1,a_2,…,a_{n-1}$ which doesn't divide $a_{n-2020}...a_{n-2}a_{n-1}$ . Proof that every number large enough appears in the sequence.