Let $X$ be a compact metric space and $f:X\to X$ be a continuous function.
Show that if $\omega(x,f)$ is a finite set, then there exists a periodic point $y$ in $X$ such that: $d(f^n(x),f^n(y))\to 0$.
I can already use that there exists a periodic point $y$ such that $\omega(x,f)=o(y,f)$ where $o(y,f)$ denotes the orbit of $y$ under $f$.
But from here I don't know what else to do, because I want the orbits of $y$ and $x$ to approach at the same time, so I don't know how to proceed.
Thanks for your help!!