Separating hyperplanes of a convex cone Let $W$ be a fixed matrix.
Define
$posW \triangleq \{t |Wy =t , y≥ 0\}$,
It is called the positive hull of $W$. It represents
the set of right-hand sides that can be obtained by a non-negative combination
of the columns of $W$ . The positive hull is easily seen to be a convex cone.
Let $p$ be a point not in the set $posW$. Then, there exist a hyperplane
$H \triangleq \{x |\sigma^Tx =0\}$ that separates $p$ and $posW$.
How can we prove that the number of possible separating hyperplanes (separating $p$  and $posW$)
is finite based on the fact that $posW$ is finitely generated?

The result is wrong in general. I.e. for $W=0$, $\text{pos }W= \{0\}$. And if $p \neq 0$, there is an infinite number of hyperplanes separating $p$ and $\text{pos } W$.
