Let $X_n$ be a Markov chain on a countable state space, $\mathbb{S}$. Let $N_n(x) = \sum_{k=1}^n\mathbb{1}_{\{X_k=x\}}$ denote the number of times the chain visits state $x\in \mathbb{S}$.
Let $x,y,z$ be recurrent states, and denote $\mu_z(y) = \mathbb{E}_z N_{\tau_z}(y)$ where $\tau_z = \inf(n \, \vert \, X_n = z)$.
Show that $\lim_{n\rightarrow \infty} \frac{N_n(y)}{N_n(z)} = \mu_z(y)$.
From Durret's book, I know that $\mu_z(y)$ represents a stationary measure for the chain ,but I'm having a hard time getting started. Any help?