Early exercise of puts

(a) Show that a European put on a non-dividend-paying stock can be worth less than its intrinsic value. Give an explicit condition on $S$ under which $p<K-S$ for every model. (b) Deduce that an American put is worth strictly more than the European put for such $S$, if $r>0$. (c) Why does the argument of Proposition 8.2.4 fail for puts?