Irreducible Markov chain and finite sets

Question

Let $(X_n)_{n\geq 0}$ be a irreducible Markov chain defined on a countable state space $S.$ Let $F \subset S$ a finite set and $\tau=inf\{n \geq 1; X_n \notin F\}$. If $x \in F$ how to prove that $\mathbb{P}_x[\tau < \infty]=1.$ I tried to use that a Markov chain defined on a closed finite class is recurrent.

Answer 1

With some help I figure out how to prove it.

First of all, using irreducibility one can prove that there is constants $a>0$ and $n \geq 1$ such that $\mathbb{P}_x[\tau>n]\leq 1-a$ for every $x \in F$. It is easy to conclude that indeed $$\mathbb{P}_x[\tau>n]\leq 1-a, \forall x \in S.$$ Now, applying the Markov property, we can prove that $$\mathbb{P}_x[\tau>kn]\leq (1-a)^k, \forall k \in \mathbb{N}.$$ The result follows easily from the last claim.