Fundamental Theorem of Asset Pricing (Linear Algebra)

I saw this question in a textbook that I was recently reading and don't really know how to aprpoach this problem.

Let $H$ be a finite dimensional vector space with inner product ($\cdotp$, $\cdotp$). Suppose $C\subset H$ is a closed convex cone such that $C\cap(-C) = \{0\}.$ Then there exists a nonzero vector p such that ($p \cdot x$) $>0$ for all nonzero $x \in C$. The book suggests to use the proof of the fundamental theorem of asset pricing and make a substitution. Could anyone let me know how to answer this problem?

(Edit: This is the theorem in the textbook with the definition of present day value.