Do the polynomials that define projective hypersurfaces whose intersection has the correct dimension necessarily form a regular sequence?

Question

Let $k$ be an algebraically closed field and let $f_1, \dots, f_n \in k[x_0, \dots, x_n]$ be homogeneous polynomials such that $V(f_1, \dots, f_n) \subset \mathbb P_k^n$ is a finite set of points. Does it automatically follow that $f_1, \dots, f_n$ is a regular sequence in $k[x_0, \dots, x_n]$, i.e., no $f_{r+1}$ is a zero divisor in the ideal generated by $f_1, \dots, f_r$? If so, why?