# Irreducible, finite Markov chains are positive recurrent

I am under the impression that an irreducible, finite Markov chain is necessarily positive recurrent. How might I show this?

Regards, Jon

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Here is one way to look at it. If $x$ is a null state, then the chain spends very little time in $x$, more precisely, $${1\over n}\sum_{j=1}^n 1_{[X_j=x]}\to 0 \text{ almost surely.}$$ Therefore, for any finite set $F$ of null states we also have $${1\over n}\sum_{j=1}^n 1_{[X_j\in F]}\to 0 \text{ almost surely.}$$