Let $(X_n)_{n\in \Bbb{N}_0}$ a irreducible markov chain on a countable state space $S$. We assume there exists a finite subset $A \subseteq S$, such that for $\tau_A := \inf\{ n \in \Bbb{N}_0 : X_n \in A\}$ it holds, that $\Bbb{P}_x(\tau_A < \infty) = 1$ for all $x\in S$.
My Question is: Is it true, that these conditions imply recurrence of the chain?
My intuition says yes, because, if the chain starts in $x$, every time the chain returns to A there is (because A is finite) a strictly positive probability to hit $x$ afterwards in finitly many steps due to irreduciblity of the chain. But I can't accomplish this rigorously.
Finally, I think I found a solution (in fact two ways to proof this). I will present the more elegant one.
We define $\tau_A^k := \inf\{n> \tau_A^{k-1} : X_n \in A\}$ recursively with $\tau_A^0 := 0$. Now we have, due to the strong markov property, that $(X_{\tau_A^k})_{k\in \Bbb{N}}$ is a markov chain. Because of our assumptions, this chain is irreducible on finite state space $A$. So, with standard theory, it has to be recurrent. We conclude that $(X_n)_{n\in\Bbb{N}}$ has this property, too.