I am trying to solve the following exercise from the book by Peterson
Let $(\Omega,\mathscr{A},\mu,T)$ be a measure preserving system. If $T^n$ is ergodic for all $n\in \mathbb Z$, and if there is a constant $c>0$ suh that $$ \limsup\limits_{n \rightarrow \infty} \mu(T^nA \cap B) \leq c\mu(A)\mu(B) $$ for all measurable sets $A$ and $B$. Then $T$ is strongly mixing, i.e. $$ \lim_{n\rightarrow\infty} \mu(T^nA \cap B) = \mu(A)\mu(B).$$
Now since $T$ is ergodic I know that $$ \frac{1}{n} \sum_{k=0}^{n}\mu(T^kA \cap B) \rightarrow \mu(A)\mu(B)$$ so if I manage to prove that $\lim \mu(T^nA \cap B) $ exists then I would be done, since the limit has to converge to the average in that case. Somebody suggested me to use Tauberian theorem, but I can't see how that will be useful.