Let $(\Omega, \mathcal{F} , P)$ be a probability space and let $T$ and $S$ be ergodic, measure-preserving transformations of $(\Omega, \mathcal{F} , P)$. Let $X : \Omega → \mathbb{R}$ be a bounded random variable such that $E[X] = 0$. Show that for every $\epsilon > 0$, there exists a positive integer $n_0 = n_0(\epsilon)$ such that $P(A_\epsilon,n_0) \leq \epsilon$, where $$A_{\epsilon,n_0}:= \left\{\sup_{n \geq n_0}\left|\frac{1}{n} \sum_{j=0}^{n-1}X\circ T^{j} - E(X)\right| > \epsilon\right\} .$$
My approach: $X \in L^{1}$, so $S_n := \frac{1}{n} \sum_{j=0}^{n-1}X◦T^{j}$ converges almost surely(hence in probability too) and in $L^{1}$ to $E(X)$, i.e., given $\epsilon > 0$, $P(|S_n - E(X)| > \epsilon) \rightarrow 0$ as $n \rightarrow \infty$,
i.e., there exist $n_0(\epsilon)$ such that for all $n \geq n_0(\epsilon)$, $P(|S_n - E(X)| > \epsilon) < \epsilon$.
$P(A_{\epsilon,n_0}) \leq \sum_{n \geq n_0} P(|S_n - E(X)|> \epsilon)$. How do I complete the proof from here.