Transitive points in CR-dynamical systems

The paper proves that if a CR-dynamical system (X,G) on a compact metric space has no isolated points and trans2(G) ≠ ∅, then (X,G) is transitive (Theorem 7.15). Its proof selects n with x_n in U and then chooses m>n with x_m in V after removing a finite prefix, leveraging that no nonempty open set can be finite when X has no isolated points. The candidate solution proves the same implication by an equivalent device: a dense forward orbit in a space without isolated points intersects every nonempty open set infinitely often, so one can pick i in U and j>i in V and conclude G^{j−i}(U)∩V ≠ ∅. Both arguments are correct and essentially the same idea, differing only in presentation (finite-prefix removal vs. infinite-returns lemma).