[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]

The theorem that [math]\mat {P}^n \to \mat {W}[/math] was proved only for the case that [math]\mat{P}[/math] has no zero entries. Fill in the details of the following extension to the case that [math]\mat{P}[/math] is regular. Since [math]\mat{P}[/math] is regular, for some [math]N, \mat {P}^N[/math] has no zeros. Thus, the proof given shows that [math]M_{nN} - m_{nN}[/math] approaches 0 as [math]n[/math] tends to infinity. However, the difference [math]M_n - m_n[/math] can never increase. (Why?) Hence, if we know that the differences obtained by looking at every [math]N[/math]th time tend to 0, then the entire sequence must also tend to 0.