Let $A$ be a commutative, regular, finitely generated $k$-algebra, where $k$ is a field of characteristic 0.

Then Grothendieck proved that in this case the algebra of differential operators on $A$ is simply the subalgebra of $End_k(A)$ generated by multiplication by $A$ and $Der_k(A)$, $k$-linear derivations. Denote this algebra by $D(A)$.

I have read that $D(A)\subseteq End_k(A)$ is 'strongly dense', i.e. given any finite-dimensional subspace $M\subseteq A$ and any linear operator $L\in End_k(A)$, there exists a differential operator $D\in D(A)$ such that $L|_M=D|_M$. Does anyone know that proof of this fact?

My thought process has been to try and work locally first: we know that $Spec(A)$ has a cover by affines $Spec(A_f)$ such that $D(A_f)\cong A_f[\frac{\partial}{\partial x_1},\dots,\frac{\partial}{\partial x_n}]$, where $x_1,\dots,x_n\in A_f$ are a system of parameters at all closed points in $Spec(A_f)$. Here the derivations $\frac{\partial}{\partial x_i}$ are only required to satisfy $x_j\mapsto \delta_{ij}$ (if I understand things correctly). So it's easy to find a differential operator which does anything you like to a subspace generated by monomials in the $x_i$, but I don't see what to do about other elements.