Is the set of subspaces generated by positive vectors a closed subset of the Grassmanian?

Question

Consider the spaces $G=\{V = (v_1|\dots|v_k)\,:\,v_i \in\mathbb{R}^n,\,\mathrm{rank}(V) = k\}$ and $G_+=\{V = (v_1|\dots|v_k)\in G\,:\,v_i \in[0,\infty)^n\}$. If $Gr(k,n)$ denotes the real Grassmanian of $k$-dimensional subspaces of $\mathbb{R}^n$, one has the quotient projection $$ \pi : G \to Gr(k,n), $$ which sends each basis $v_1,\dots,v_k$ in the subspace of $\mathbb{R}^n$ spanned by it. How can I prove that $\pi(G_+)$ is closed in $Gr(k,n)$?