A dynamical system $(X,\phi)$ is said to be topologically transitive if for any non-empty open sets $U,V\subset X$, there exists a positive integer $n$ such that $\phi^n(U)\cap V \neq \emptyset$.
Birkhoff's Transitivity theorem asserts tat if $X$ is a second countable, complete metric space, then topological transitivity implies that there is a dense set of points in $X$ with dense orbit.
In theorem 2.5 here, the proof goes that $\cup_{m=1}^\infty \phi^{-m}(U_k)$ is dense, by topological transitivity. I cannot see why this claim is true. Can anyone help?