Let $f: X \to X$ be a homeomorphism from a compact metric space $X$ to itself.

$x$ is said to be a non-wandering point of $f$ if for all open neighborhood of $x$ say $U$ there exists $n \in \mathbb{Z}$ such that $f^n(U) \cap U \neq \emptyset$. The set of all non-wandering points of $f$ is denoted by $\Omega(f)$. I'm trying to prove the following exercise :

Show that if $ x \in \Omega(f)$ and $U$ is a neighborhood of $x$, there is a sequence of integers $n_i$ tending to infinity such that the intersection $f^{n_i}(U) \cap U$ is non-empty.

There is a proposition in the book which says if $U$ is a neighborhood of $\Omega(f)$ then for every $x \in X$ there is $ N>0$ such that for every $n \geq N $ , $f^n(x) \in U$.

May I use the proposition above to prove the main problem?


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