If $y_0\in\langle x_0\rangle$, from some orbit $\langle x_0\rangle$, then the eventual behavior of $\langle y_0\rangle$ is the same as the eventual behavior of $\langle x_0\rangle$. In fact, if $y_0=x_k$ for some $k\in\mathbb{N}$, then $\langle y_0\rangle = \langle x_k\rangle$. Explain why this is so.
I understand this intuitively where $\langle x_0\rangle = \langle x_0,x_1,x_2,...,y_0,y_1,y_2,...\rangle$ and so $\langle y_0\rangle$ follows the behavior of $\langle x_0\rangle$.
But is there a more technical way to explain this?