Corollary

— For any subset A of E, the convex cone generated by A, is identical with the set of linear combinations $\sum_{i \in J} \lambda_i x_i$, where $(x_i)_{i \in J}$ is any finite non-empty family of points of A, and where $\lambda_i > 0$ for all $i \in J$.

It is sufficient to see that, if a convex cone contains a point $x \neq 0$ of A then it also contains the half-line $C_x$ of the points $\lambda x$ where $\lambda$ varies in the set of positive numbers and that $C_x$ is a convex cone.