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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?

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