MEMO 2024 T4

A finite sequence $x_1, x_2, \ldots, x_r$ of positive integers is a palindrome if $x_i = x_{r+1-i}$ for all integers $1 \leq i \leq r$.

Let $a_1, a_2, \ldots$ be an infinite sequence of positive integers. For a positive integer $j \geq 2$, denote by $a[j]$ the finite subsequence $a_1, a_2, \ldots, a_{j-1}$. Suppose that there exists a strictly increasing infinite sequence $b_1, b_2, \ldots$ of positive integers such that for every positive integer $n$, the subsequence $a[b_n]$ is a palindrome and $b_{n+2} \leq b_{n+1} + b_n$. Prove that there exists a positive integer $T$ such that $a_i = a_{i+T}$ for every positive integer $i$.