Why $\{T_y<\infty, (X_{T_y+n})_n \text{ do not visit } x\}\subset \{N_x<\infty\}$?

Question

Let $(X_n)$ be a Markov chain with values in some countable set $S$ and assume $x\rightarrow y$, i.e. $\Bbb{P}_x(T_y<\infty)>0$ where $T_y=\inf \{n\geq 1: X_n=y\}$. Now assume further that $\Bbb{P}_y(T_x<\infty)<1$. Then why do we have $$\{T_y<\infty, (X_{T_y+n})_n \text{ do not visit } x\}\subset \{N_x<\infty\},$$ Where $N_x$ is the number of visits of $x$ by the chain?

I somehow don't see why this should be true and how to get to this. We used this the get an inequality of probability in order to apply the Markov property later. But could someone show me why this is true and maybe give some intuition how one could get such an inequality so that we can apply the Markov property later?

Answer 1

The event on the left-hand side is "state $y$ was visited, and moreover, state $x$ is never visited after the first time state $y$ was visited." This implies that the number of visits to $x$ is finite.

Answer 2

Here is a classical result in the theory of Markov chain, very useful to give a classification of the states: if a state $y$ can is reachable from a recurrent state $x$, then it is also recurrent and both states communicate. This answers your question.