= Solution
Interpret positivity of $Y$ as strict positivity almost surely. Suppose the stated bounded positive pricing variable exists. For a vector $H$ satisfying the two sign restrictions,
$$
0\geq H\cdot p=\mathbb E[Y(H\cdot P)]\geq0.
$$
A nonnegative random variable with zero expectation vanishes almost surely. Since $Y>0$, this gives $H\cdot P=0$ almost surely and $H\cdot p=0$. The supplied nondegeneracy condition now yields $\boxed{H=0}$. This proves the direction from a <one-period martingale deflator> to the separating-portfolio condition; no asset-pricing theorem has been invoked.
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