Let $\mathbb{P}^n$ be a projective space over an algebraic closed field $k$. I want to show that any finitely many points in $\mathbb{P}^n$ can be included in an affine open chart of it. A candidate would be the principal open set i.e. the complement of a hypersurface $\mathbb{P}^n-V(f)$ which is affine. However, I have trouble constructing this homogeneous polynomial $f$ i.e. given finitely many points $a_0, a_1,...\in \mathbb{P}^n$ then $a_0,a_1,...\not\in V(f)$. The only thought I have is to look at them on an affine chart and I can construct an $f'$ in affine space, and then homogenize it. However, this will make me circular i.e. I come back to find an affine chart that includes those points. So I guess this won't work.
Appreciate any ideas or hints!
A fairly direct way to see this is via duality: we search for a hyperplane $H \subseteq \mathbb{P}^n_k$ such that $a_1,\dots,a_m \notin H$ - the complement of $H$ will thus contain $a_1,\dots,a_m$ and be isomorphic to $\mathbb{A}^n_k$.
If, by contradiction, every hyperplane $H$ contained $a_i$ for some $i$, then - passing to ${\mathbb{P}^n_k}^\ast \cong \mathbb{P}_k^n$ - we get that every point in $\mathbb{P}^n_k$ is contained in at least one of the $m$-many hyperplanes corresponding to $a_1,\dots,a_m \implies \mathbb{P}^n_k$ is a union of $m$ hyperplanes, which of course contradicts the fact the $\mathbb{P}^n_k$ has dimension $n > n-1$.