Let $\xi_1, \ldots, \xi_k$, $k$ elements of $\mathbb{R}^d$ and
$$F := \left\{\sum_{i=1}^k \lambda_i \xi_i, (\lambda_1, \ldots, \lambda_k) \in \mathbb{R}_+^k\right\}$$
show that $F$ is a closed convex cone. Let $\xi \in \mathbb{R}^d$, show the equivalence between:
\begin{itemize}
\item $\xi \in F$,
\item $\xi \cdot x \geq 0$ for all $x \in \mathbb{R}^d$ such that $\xi_i \cdot x \geq 0, i = 1, \ldots, k$.
\end{itemize}