Lemma 1

— If a cone $A$, with vertex $0$, is contained in an open half-space determined by a hyperplane $H$, then it is contained in a closed half-space determined by a hyperplane $H_0$, that is parallel to $H$ and passes through $0$.

Let $f(z) = \alpha$ with $\alpha < 0$ be an equation of $H$, so that $f(z) = 0$ is the equation of $H_0$. If there exists $z \in A$ such that $f(z) < 0$, then there would exist $\lambda > 0$, such that $f(\lambda z) = \alpha$, and as $\lambda z \in A$, this would contradict the hypothesis.