Let $(X_n)$ be a finite Markov chain. $R_i$, $i = 1,\dots, l$ are the recurrent classes with invariant probability $\pi_i$. For all $i \in 1,\dots, l$ define $\tau_i = \inf \{n\geq 1 | X_n\in R_i\}$. $f$ is a bounded function.
Then $\mathbb{P}$-as $\frac{1}{n}\sum_{k=0}^n f(X_k) \to \sum_{i=1}^l \mathbb{1}(\tau_i<\infty)\int fd\pi_i$ as $n\to +\infty$.
I cannot find a proof of this theorem, can someone give a reference or a demonstration?