Theorem. If C is a finite closed and irreducible set, then all states in C are recurrent.
-Essentials of Stochastic Processes, Rick Durrett.
I just need to set the next implications straight since literature can be very confusing when talking about finite state space Markov chains.
If is an irreducible chain with finite state space then
even more, is positive recurrent?
I have found out the answer to both questions is positive but i would like to confirm it, is anybody sure about it? Can you think of any text where it explicitly shows the above results?