 Arithmetic-progression multiple averages under weak mixing

For bounded observables of a probability weak mixing system, the displayed convergence holds in $L^2$. Induct on $k$: center the final observable, compute the lagged Hilbert-space correlations, apply the induction hypothesis to the $k-1$ remaining factors, and use weak mixing of $T^k$ with the Van der Corput lemma to obtain zero. Adding back the final mean completes the induction.