[math] \newcommand{\NA}{{\rm NA}} \newcommand{\mat}[1]{{\bf#1}} \newcommand{\exref}[1]{\ref{##1}} \newcommand{\secstoprocess}{\all} \newcommand{\NA}{{\rm NA}} \newcommand{\mathds}{\mathbb}[/math]

Let [math]\mat{P}[/math] be the transition matrix of an [math]r[/math]-state ergodic chain. Prove that, if the diagonal entries [math]p_{ii}[/math] are positive, then the chain is regular.