1. Question about irreducibility ,recurrence of a Finite M. Chain

hi!

I just need to set the next implications straight since literature can be very confusing when talking about finite state space Markov chains.

If $X_n$ is an irreducible chain with finite state space then

is $X_n$ recurrent?

even more, is $X_n$ 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?

Thank you!

2. Theorem. If C is a finite closed and irreducible set, then all states in C are recurrent.

-Essentials of Stochastic Processes, Rick Durrett.

3. thank you, but what about positive recurrence?