lemma on projection of countable sets

Suppose $𝔽$ is an infinite field and $S$ is an infinite subset of${𝔽}^n$. Then there exists a line $L$ such that the projection of $S$on $L$ is infinite.

Proof: This proof will proceed by an induction on $n$. The case$n=1$ is trivial since a one-dimensional linear space is a line.

Consider two cases:

Case I: There exists a proper subspace of ${𝔽}^n$which contains an infinite number of points of $S$.

In this case, we can restrict attention to this subspace. By theinduction hypothesis, there exists a line in the subspace such thatthe projection of points in the subspace to this line is alreadyinfinite.

Case II: Every proper subspace of ${𝔽}^n$ contains at most afinite number of points of $S$.

In this case, any line will do. By definition, one constructs aprojection by dropping hyperplanes perpendicular to the line passingthrough the points of the set. Since each of these hyperplanes willcontain a finite number of elements of $S$, an infinite number ofhyperplanes will be needed to contain all the points of $S$, hence theprojection will be infinite.