 Eventually periodic sequence

A sequence is eventually periodic if there are integers $N\ge 0$ and $T>0$ such that $u_{n+T}=u_n$ for all $n\ge N$. 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.