 Solution

Apply part b with initial capital ${\xi }_0+\epsilon$ and let $\epsilon \downarrow 0$. The finite-dimensional attainable space is closed, so a limit portfolio $H$ satisfies $H\cdot P_0\le {\xi }_0$ and $H\cdot P_1\ge {\xi }_1$. Let $\bar{Y}$ be a deflator for which equality holds. ThenStrict positivity of ${\bar{Y}}_1$ forces $H\cdot P_1={\xi }_1$ almost surely, and the middle identity then gives $H\cdot P_0={\xi }_0$.