An eventually periodic point must be an asymptotically periodic point?

Question

Most definition can be found from Eventually periodic point and homeomorphism. And you can also find the definitions from the classic paper "Period Three Implies Chaos", my question is in the caption, that is "An eventually periodic point must be an asymptotically periodic point?", an asymptotically periodic point x means that you can find a periodic point p such that the distance from point x to p irtated from the continuous maps f is zero.

Answer 1

Suppose that $x$ is an eventually periodic point for the map $f$. Then there exists $k \ge 1$ such that $y= f^k(x)$ is a periodic point of period $l \ge 1$.

The corresponding periodic orbit is then $\{y, f(y), \ldots, f^{l-1}(y) \}$. Choose a point $p$ in this orbit such that $f^k(p) = y$. Then $p$ is a periodic point of $f$ and for $n \ge k$, we have $f^n(x) = f^n(p)$ which implies that $x$ is asymptotically periodic.