Suppose we have a discrete-time Markov chain with $n$ states, with transition matrix $P$. The definition of irreducibility tells us that for any two states $x$ and $y$, there exists an integer $t$ such that $P^t (x,y) >0$.
But how does this imply that the states can be labelled as $s_1 , s_2 , \ldots, s_n$ such that
$\quad \quad \quad \quad \quad \quad \quad \quad \quad $ $P(s_{i}, s_{i+1}) >0, \quad \quad $ for all $ \quad i \in \{1, \ldots, n-1 \}$ ?
Any suggestions of how to show this by definition?
This statement comes from a proof from Levin's book of Markov Chains:
It doesn't. That would mean that every connected graph has a Hamiltonian path. That's not the case; consider a star graph.