A finite sequence $x_1,\dots,x_r$ of positive integers is a palindrome if $x_i=x_{r+1-i}$ for all integers $1 \le i \le r$. Let $a_1,a_2,\dots$ be an infinite sequence of positive integers. For a positive integer $j \ge 2$, denote by $a[j]$ the finite subsequence $a_1,a_2,\dots,a_{j-1}$. Suppose that there exists a strictly increasing infinite sequence $b_1,b_2,\dots$ of positive integers such that for every positive integer $n$, the subsequence $a[b_n]$ is a palindrome and $b_{n+2} \le b_{n+1}+b_n$. Prove that there exists a positive integer $T$ such that $a_i=a_{i+T}$ for every positive integer $i$.