Square-correlation proof of weak mixing from product ergodicity ($\frac{1}{N}{\sum }_{n<N}|⟨U^{n}f,g⟩{|}^2\to 0$)

For mean-zero $f$, apply the mean ergodic theorem on an ergodic product system to $f\otimes \overline{f}$, paired with $g\otimes \overline{g}$. Its correlation is $|⟨U^{n}f,g⟩{|}^2$. The Cauchy-Schwarz inequality then gives averaged absolute correlation decay, the definition of weak mixing.