Let $X$ be an irreducible Markov chain with countable state space $S$ and transition matrix $\Pi$. For $x\in S$, let $D_x:=\{n:\Pi^n(x,x)>0\}$ and let $d_x$ be the greatest common divisor (gcd) of $D_x$.
The part I don't get is when the proof introduces $m$. How do we know there exists $md_x$ and $(m+1)d_x$. What is the main idea behind to show that $D_x$ contains all sufficiently large multiplies of $d_x$?