The proof in the solutions manual seems to rely heavily on the Kolmogorov's Zero-One law, beginning as follows:
Since $S$ is finite, there is at least one $i_0 \in S$ such that for infinite times $X_n = i_0$. Hence, $P(X_n = i_0\ i.o.) = 1$, and consequently $P_{i_0}(X_n = i_0\ i.o.) > 0$.
Later it also explicitly applies the Zero-One law to the event $\{X_n = i_0\ i.o.\}$.
The problem I'm having is that I'm not sure how the tail field should look like in this particular case. What events generate it?
Intuitively, the events should be something along the lines of $A_n = \{X_n = i_0\}$, but in this case it's fairly obvious the events are not independent, and I'm not sure how to "disentangle" them by taking just the suitable subsequence of the events.