I am trying to show that for any set, $A\neq \{ 0\}$ and any $x\in co(cone A) \subset \mathbb{R^n}$, there exists linearly independent set $A' \subset A$ such that $x\in co (cone A')$.
Here, co(A) means convex hull of set A, and cone(A) means the cone generated by a set A.
I think some how I have to use
co(cone A) = cone(co A)=$\{ \sum ^{k} _{i=1} \lambda_i x^{(i)} | k\in \mathbb{N}, \lambda_i \geq0, x^{(i)}\in A\}$, where $x^{(i)}$ is a vector in $A$
But, I am not sure why and where we need linear independence here and how to sue the linear independency. Please give me some advice.