In Hartshorne's Algebraic Geometry Exercise I.4.8,
Exercise: Show that any variety of positive dimension over $k $ has the same cardinality as $ k $.
Hints: Do $\mathbb{A}^{n} $ and $\mathbb{P}^{n}$ first. Then for any $ X $, use induction on the dimension $ n $. Use (4.9) to make $ X $ birational to a hypersurface $ H \subset\mathbb{P}^{n+1 }$. Use (Ex. 3.7) to show that the projection of $ H$ to $\mathbb{P}^{n}$ from a point not on $ H$ is finite-to-one and surjective.
Although this has been asked in many posts in MSE, namely:Cardinality of variety and Cardinality of quasiaffine variety . My question is different. In these posts, it seems that the proof provided (although they are indeed wonderful and helpful) did not follow the hints by Hartshorne. I tried to follow the hints, but didn't get the promising results.
So my question is: How to finish this exercise using the methods provided in the HINTS?
My attempts:
(1) I had made some attempts and found that using Noether normalisation theorem, we may direct prove the desired result. Maybe I will post an answer using this in the posts mentioned above. Yet I still hope to know how to follow the hint.
(2) I have managed to prove that the cardinality of $\mathbb{A}^{n} $ and $\mathbb{P}^{n}$ are the same as the cardinality of the ground algebraically closed field $k$. Yet what's next?