$\newcommand{\Pj}{\mathbb{P}}$I am trying to show that if $k$ is an infinite field, then for any $r$-dimensional closed subset $X\subset\Pj^n_k$ there are $r$ hyperplanes (i.e., $V(f_i)$ for linear homogeneous polynomials $f_i\in k[x_0,\dots,x_n]$) such that $X\cap\bigcap_{i=1}^r V(f_i)$ consists of finitely many points. (This is essentially exercise 12.3.C(d) in the latest version of Vakil’s notes.)
I have already solved the previous parts of the problem, with the exception of the very last part of (c), which also involves the infinite field hypothesis. I am thinking it would be useful to try to imitate my argument from the earlier parts of this problem (which more or less matches the solution given in this answer, for reference: Trouble with Vakil's FOAG exercise 11.3.C).
But I don’t see how we can use the hypothesis that $k$ is infinite to guarantee (1) that the $f_i$ are linear and (2) that the intersection has only finitely many points. I am aware of some vague intuition that over finite fields you can have curves that hit every rational point, and that the infinite field hypothesis is probably meant to prevent this, but I don’t understand how this will play out algebraically in the proof. The only time I remember seeing such a hypothesis before was in a statement of the prime avoidance lemma (where it comes into play because no vector space over an infinite field is a union of finitely many proper subspaces), and I have tried to leverage that here, but I haven’t had any luck.
Any hints, references, or solutions addressing (1) and (2) above would be much appreciated.