## Markov Chain

Let $\Pi$ be the transition matrix for an irreducible Markov chain on a state Space S. If $\alpha$ is a stationary distribution for $\Pi$ .

Prove that if S is finite, $\alpha(x) > 0$ for all $x$.

I would appreciate some hints/useful definitions as I'm not really getting Markov chains (and am generally rubbish at probability!).

Thanks.