 Solution

The one-period fundamental theorem of asset pricing, applied as a separating-hyperplane theorem to the cone of attainable payoffs, gives the superhedging dualitywhere the supremum is over martingale deflators. The assumed strict inequalities make the right side strictly below ${\xi }_0$. Hence some $H$ satisfies $H\cdot P_0\le {\xi }_0$ and $H\cdot P_1\ge {\xi }_1$ almost surely.