Question 88
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 $a\left[b_{n}\right]$ 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 $i$.
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 $a\left[b_{n}\right]$ 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 $i$.