lean_workbook_plus_20472

Prove that there exist two infinite sequences $(A_n)_{n\ge1}$ and $(B_n)_{n\ge1}$ of positive integers such that the following conditions hold simultaneously:\n\n(i) $1 < a_1 < a_2 < a_3 < \ldots$\n\n(ii) $a_n < b_n < a_n^2$, for all $n \ge 1$\n\n(iii) $a_n - 1$ divides $b_n - 1$, for all $n \ge 1$\n\n(iv) $(a_n)^2 - 1$ divides $(b_n)^2 - 1$, for all $n \ge 1$