Show that if dynamical system $(X,T)$ is weakly-mixing, then $(X, T^n)$ is weakly mixing.
I am using this definition of weakly mixing: we say that system is weakly mixing if system $(X^2, T \times T$) is transitive.
So now I know that for every nonempty, open sets $U_1, V_1, U_2, V_2 ⊂ X $ there exists $m ∈ N$ such that $T^m(U_1\times U_2)∩(V_1 \times V_2) \ne ∅$.
I need to show that $(X^2, T^n \times T^n)$ is transitive, so that for every nonempty, open sets $U_1, V_1, U_2, V_2 ⊂ X $ there exists $k ∈ N$ such that $T^{nk}(U_1\times U_2)∩(V_1 \times V_2) \ne ∅$.
I lack experience in dynamical systems, so any help would be appreciated.