Let $\sigma$ be a convex polyhedral cone and $V$ be span ($\sigma$). Then associated to any facet $\tau$ of $\sigma$ there is a unique $u_\tau$ up to scalar multiplication such that $Ann \, u_\tau \cap \sigma=\tau$(for otherwise $codim(span \tau \hookrightarrow V)\neq 1$). Let $H_\tau$ be the 'supporting hyperplane' on which $u_\tau$ is at least 0.
Using the fact that the topological boundary is the union of all the faces and and thus the facets of $\sigma$, Fulton proves that $\cap_{\tau \text{ is a facet}} H_\tau=\sigma$.
I want to ask about how to complete on of the steps in the following proof I did myself about this before I saw how Fulton did it.
Here is the outline, the steps of which I know are correct a postiori.
Suppose $v \in \cap_{\tau \text{ is a facet}} H_\tau$. I know ($ u_\tau(v)\geq 0 \,\,\forall \tau$ a facet $\Rightarrow \lambda(v) \geq 0 \,\,\forall \lambda $ defining a face contained in $\tau$). It then follows from the fact that every face is contained in a facet and $\check{\check \sigma}=\sigma$ that $ v \in \sigma$.
My question: How can I show the step ($ u_\tau(v)\geq 0 \,\,\forall \tau$ a facet $\Rightarrow \lambda(v) \geq 0 \,\,\forall \lambda $ defining a face contained in $\tau$)?