Suppose that a sequence {$a_n$} is eventually periodic. Prove that {$b_n$} is eventually periodic, given that they are in some relation.

Question

1. To clarify, a sequence {$x_n$} is called eventually periodic if there exist positive integer $r$ and $p$ for which $x_{n+p}=x_n$ for all $n\geq r$.

2. The relation they have is that,

   (i). $b_{n+1}=b_n+a_{n+1}$, if $b_n\leq a_{n+1}$

   (ii). $b_{n+1}=b_n-a_{n+1}$, if $b_n> a_{n+1}$