Let $(X_n)_{n\in \mathcal{N_0}}$ an i.i.d. sequence on a generic $E$, $\tau$ be the measure preserving shift operator and suppose there exist, for $A,B\in\mathcal{A}$,the product sigma algebra, $A^\epsilon$ and $B^\epsilon$ such that $P[A\triangle A^{\epsilon}]<\epsilon$ and $P[B\triangle B^{\epsilon}]<\epsilon$ for every $\epsilon>0$.
Suppose also that for big $n$, $A^{\epsilon}$ and $\tau^{-n}(B^\epsilon)$ are independent.
In Example 20.26 of Probability Theory by A. Klenke (3rd version), it is shown that
$$ \limsup_{n\to\infty}|P[A\cap \tau^{-n}(B)]-P[A]P[\tau^{-n}(B)]|\leq \limsup_{n\to\infty}|P[A^\epsilon\cap \tau^{-n}(B^\epsilon)]-P[A^\epsilon]P[\tau^{-n}(B^\epsilon)]|+4\epsilon $$
and so the author concludes that the process is mixing.
How can I get to this bound? I do not understand how to obtain the $4\epsilon$ term.
Please, let me know if more context is needed. Thanks for the help.
Using \begin{align} \left\lvert \mathbf{1}_{E\cap F}-\mathbf{1}_{E'\cap F'}\right\rvert &=\left\lvert \mathbf{1}_{E}\mathbf{1}_{F}-\mathbf{1}_{E'}\mathbf{1}_{F'}\right\rvert\\ &=\left\lvert \mathbf{1}_{E}\left(\mathbf{1}_{F}-\mathbf{1}_{F'}\right)+\left(\mathbf{1}_{E}-\mathbf{1}_{E'}\right)\mathbf{1}_{F'}\right\rvert\\ &\leqslant \mathbf{1}_{F\Delta F'}+\mathbf{1}_{E\Delta E'}, \end{align} we derive that $$ \left\vert\mathbb P(E\cap F)-\mathbb P(E'\cap F')\right\rvert \leqslant \mathbb P(E\Delta E')+\mathbb P(F\Delta F'). $$ I guess that $\tau$ is measure preserving hence applying the previous inequality to $E=A$, $F=\tau^{-n}B$, $E'=A^\varepsilon$ and $F'=\tau^{-n}B^\varepsilon$, we get that $$ \mathbb P(A\cap \tau^{-n}B)\leqslant 2\varepsilon+\mathbb P(A^\varepsilon\cap \tau^{-n}B^\varepsilon). $$