= Solution
Apply part b with initial capital $\xi_0+\varepsilon$ and let $\varepsilon\downarrow0$. The finite-dimensional attainable space is closed, so a limit portfolio $H$ satisfies $H\cdot P_0\leq\xi_0$ and $H\cdot P_1\geq\xi_1$. Let $\bar Y$ be a deflator for which equality holds. Then
$$
0\leq\mathbb E[\bar Y_1(H\cdot P_1-\xi_1)]
=\bar Y_0(H\cdot P_0-\xi_0)\leq0.
$$
Strict positivity of $\bar Y_1$ forces $H\cdot P_1=\xi_1$ almost surely, and the middle identity then gives $H\cdot P_0=\xi_0$.
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