# lemma on projection of countable sets

Suppose $\mathbb{F}$ is an infinite field and $S$ is an infinite subset of $\mathbb{F}^{n}$. 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 $\mathbb{F}^{n}$ which contains an infinite number of points of $S$.

In this case, we can restrict attention to this subspace. By the induction hypothesis, there exists a line in the subspace such that the projection of points in the subspace to this line is already infinite.

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

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

Title lemma on projection of countable sets LemmaOnProjectionOfCountableSets 2013-03-22 15:44:53 2013-03-22 15:44:53 rspuzio (6075) rspuzio (6075) 10 rspuzio (6075) Theorem msc 14A10