= Solution
Suppose an equivalent measure $Q$ made $S$ a local martingale. Stop at the positive time furnished by part (i), chosen so that $S_{t\wedge\tau}\geq0$. A <nonnegative local martingale> is a supermartingale, and this one starts from $S_0=0$. Hence $S_{t\wedge\tau}=0$ $Q$-almost surely for every deterministic $t$.
But $Q$ and $P$ have the same null events, while $\tau>0$ almost surely. Some positive rational $q$ therefore satisfies $Q(q<\tau)>0$, and on that event part (i) gives $S_q>0$, a contradiction. No such equivalent measure exists.
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