proof of Poincaré recurrence theorem 1

Let $A_n={\cup }_{k=n}^{\mathrm{\infty }}{f}^{-k}E$. Clearly, $E\subset A_0$ and$A_i\subset A_j$ when $j\le i$. Also, $A_i=f^{j-i}{A}_j$, so that $\mu (A_i)=\mu (A_j)$ for all $i,j\ge 0$, bythe $f$-invariance of $\mu$.Now for any $n>0$ we have $E-A_n\subset A_0-A_n$, so that

$$\mu (E-A_n)\le \mu (A_0-A_n)=\mu (A_0)-\mu (A_n)=0.$$

Hence $\mu (E-A_n)=0$ for all $n>0$, so that$\mu (E-{\cap }_{n=1}^{\mathrm{\infty }}{A}_n)=\mu ({\cup }_{n=1}^{\mathrm{\infty }}E-A_n)=0$.But $E-{\cap }_{n=1}^{\mathrm{\infty }}{A}_n$ is precisely the set of those $x\in E$ suchthat for some $n$ and for all $k>n$ we have $f^k(x)\notin E$. $\mathrm{\square }$