How does one prove that
(1) a minimal $\mathbb{Z}$-action on an infinite compact Hausdorff space is free?
(2) for such an action, we can find a nonempty open subset $U$ of the space such that $nU\cap U=\emptyset$ for, say, all $n\in[-5N,5N]\setminus\{0\}$?
For convenience let me write the action as $\phi^n : X \to X$ for each $n \in \mathbb{Z}$, where $\phi : X \to X$ is some homeomorphism.
For (1), if the action is not free then there exists $n \ne 0$ and $x \in X$ such that $\phi^n(x)=x$. The finite set $\{x,\phi(x),...,\phi^{n-1}(x)\}$ is therefore an invariant subset. Finite subsets being closed and proper, this violates the assumption that the action is minimal.
For (2), first pick pairwise disjoint open subsets $U_n$ such that $\phi^n(x) \in U_n$, $-5N \le n \le 5N$. Then let $$U = \bigcap_{n=-5N}^{5N} \phi^{-n}(U_n) $$