# 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 |