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 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 $\\mathbb{F}^{n}$ 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.

单词	LemmaOnProjectionOfCountableSets
释义	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 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 $\\mathbb{F}^{n}$ 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.
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