In The homogeneous ideal of $2n$ points in general position in $\mathbb{P}^n$, we let $\Gamma$ be a set of $d=2n$ points in general position in $\mathbb{P}^n$, and we want to show that the associated homogeneous ideal $I(X)$ is generated by homogeneous polynomials of degree $2$. If $\Gamma$ is a set of $d\leq kn$ points of $\mathbb{P}^n$ in general position, where $k\geq 2$, how could I show that $\Gamma$ vanishes for a family of homogeneous polynomials of degree $\leq k$?
The main problem I have is that I want to prove the result using Hilbert polynomials and resolutions only, my professor told me there is a very easy way to prove it using such tools but I don't see how.