In the setting of Proposition 4.27, show that, if, for any \(i=1, \ldots, I\), the utility function \(u^{i}: \mathbb{R}^{2} \rightarrow \mathbb{R}\) is non-decreasing in the first argument and strictly increasing in the second argument and there exists a portfolio \(\hat{z} \in \mathbb{R}^{N}\) such that \(D \hat{z}>0\), then the existence of an optimal portfolio excludes the existence of arbitrage opportunities.