Nemotron-IMO-Bench 2026 A15

The first derivative of an infinite sequence $(a_n)_{n\geq 1}$ is the sequence $(a'_n)_{n\geq 1}$ with $a'_n=a_{n+1}-a_n$ for $n\geq 1$. Define the $k$th derivative as the first derivative of the $(k-1)$th derivative for $k\geq 2$. Say the sequence $(a_n)_{n\geq 1}$ is good if the sequence itself and all of its derivatives consist of positive numbers. Prove that if $(a_n)_{n\geq 1}$ and $(b_n)_{n\geq 1}$ are good sequences, then so is the sequence $(a_nb_n)_{n\geq 1}$.