= Solution
A <numéraire portfolio> $\eta$ satisfies $\eta\cdot P_0>0$ and $\eta\cdot P_1>0$ almost surely. If an arbitrage $H$ already has zero initial cost, it is a terminal-consumption arbitrage. Otherwise $H\cdot P_0<0$; set
$$
\widetilde H=H-\frac{H\cdot P_0}{\eta\cdot P_0}\eta.
$$
Then $\widetilde H\cdot P_0=0$, while its terminal payoff is the nonnegative payoff of $H$ plus a strictly positive multiple of $\eta\cdot P_1$. Hence it is strictly positive almost surely and is a terminal-consumption arbitrage.
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