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 N th time tend to 0, then the entire sequence must also tend to 0.