Proposition 2 (Kronecker)

Tag 0BY6, permanent. Cite it as [Bourbaki, Tag 0BY6].

General Topology, chapter VII, § 1. Subgroups and quotient groups of $\mathbf{R}^n$

Let $\theta_1, \theta_2, \ldots, \theta_n$ be $n$ real numbers. In order that, for each $\varepsilon > 0$ , there exist an integer $q$ and $n$ integers

p_i \quad (1 \leq i \leq n)

such that

|q\theta_i - p_i| \leq \varepsilon \quad (1 \leq i \leq n),

where the left-hand side of at least one of these inequalities does not vanish, it is necessary and sufficient that at least one of the $\theta_i$ be irrational.

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Exercise 11, § 1

Cites

Nothing.