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
Canonical name	LemmaOnProjectionOfCountableSets
Date of creation	2013-03-22 15:44:53
Last modified on	2013-03-22 15:44:53
Owner	rspuzio (6075)
Last modified by	rspuzio (6075)
Numerical id	10
Author	rspuzio (6075)
Entry type	Theorem
Classification	msc 14A10