Eventually periodic sequence
= Eventually periodic sequence
{title2=$u_{n+T}=u_n\quad(n\geq N)$}
A <sequence> is eventually periodic if there are <integers> $N\geq0$ and $T>0$ such that $u_{n+T}=u_n$ for all $n\geq N$. It consists of a finite initial segment followed by a repeating period. Checking finitely many repeated values can suggest eventual periodicity, but a proof requires a reason the repetition continues, such as recurrence of a complete state in a deterministic finite-state process.