How to show that Hilbert function of points in projective space is constant for large values $\in N$? Recall that for $X\subset \mathbb{P}^{n}$ an algebraic set with homogeneous coordinate ring $Γ(X) = k[x_1, ..., x_{n+1}]/I(X)$, the Hilbert function of $X$ is a function $h_{X }: \mathbb{N} → \mathbb{N}$ defined
as: $h_{X}(m)$ = dim$_k(Γ(X)_m)$.

Let $X ⊂ P^n$ be a set of $d$ points. Show that $h_{X}(m) = d$ for $m \geq d-1$.

By explicit computation, I've checked this result for $d=1,2,3,4,...$ but i'm unable to prove this for arbitrary $d$ points. Any ideas?
 A: Fix a linear function $L$ that doesn't vanish on any point of $X$. We have a map $\phi: \Gamma(X)_m\rightarrow \prod_{p\in X}{k}$ that sends a function $f$ to the value of $\frac{f}{L^m}$ at each point $p\in X$. As said in the comment below, the purpose of $L$ is to make $\phi$ well-defined, as scaling the coordinates of $p\in X$ will scale $f(p)$. By definition of $\Gamma(X)$, this is injective, so $h_X(m)\leq d$ in general.
If we are given a value of $m\geq d-1$, and want to show $h_X(m)=d$, it suffices to show $\phi$ is surjective. To do this, it suffices to find, for each $p_{0}\in X$, a function $f\in \Gamma(X)_m$ that vanishes on $p\in X$ for $p\neq p_0$, but does not vanish on $p_0$. 
To do this, pick hyperplanes $\{L_p=0\}$ for each $p\in X$ such that $L_p$ passes through $p$ but not any other point in $X$. If $m=d-1$, we can let $f=\prod_{p\neq p_0}{L_p}$. If $m>d-1$, we can multiply $\prod_{p\neq p_0}{L_p}$ by any function not vanishing on $p_0$. 
Harris' First Course in Algebraic Geometry has a chapter which covers this and a bit more.
