Question 6
Let a sequence $(a_n)_{n\ge1}$ satisfy
$$0\lt a_{n+1}-a_n\le2001\qquad(n\ge1).$$
Prove that there are infinitely many pairs of positive integers $(p,q)$ such that $p\lt q$ and $a_p\mid a_q$.
Editorial note: the supplied page does not specify that the terms are integers. The standard divisibility statement is proved below under the explicitly stated integer-term assumption, corroborated by another transcription.
$$0\lt a_{n+1}-a_n\le2001\qquad(n\ge1).$$
Prove that there are infinitely many pairs of positive integers $(p,q)$ such that $p\lt q$ and $a_p\mid a_q$.
Editorial note: the supplied page does not specify that the terms are integers. The standard divisibility statement is proved below under the explicitly stated integer-term assumption, corroborated by another transcription.