Question: Show that all state in a finite irreducible Markov chain are recurrent.
Attempt: First I considered that a finite irreducible Markov chain is transient.
since there are only a finite number of states, it is not possible for the process to hit each state only a finite number of time. Hence a finite irreducible MC is recurrent.
countering my hypothesis I proved the question.
My question is how can I prove this mathematically? Is there any way?