The theorem that Pn→W was proved only for the case that P has no zero entries. Fill in the details of the following extension to the case that P is regular. Since P is regular, for some N, PN has no zeros. Thus, the proof given shows that MnN−mnN approaches 0 as n tends to infinity. However, the difference Mn−mn can never increase. (Why?) Hence, if we know that the differences obtained by looking at every Nth time tend to 0, then the entire sequence must also tend to 0.