TITLE: Dirichlet's approximation theorem (simultaneous version): proof via Minkowski's theorem
QUESTION [5 upvotes]: There is a proof of the Dirichlet's approximation theorem based on Minkowski's theorem. The proof is given on wikipedia (http://en.wikipedia.org/wiki/Dirichlet%27s_approximation_theorem) and it is quite simple. They claim that it naturally extends to the proof of the simultaneous version of the theorem. I don't quite see how.
Thank you for help.

REPLY [2 votes]: The volume of 
$$
S = \{ (x,y_1, \dots, y_d) \in \mathbb{R}^{1+d}; -N-\frac{1}{2} \le x \le N+\frac{1}{2}, |\alpha_i x - y_i| \le \frac{1}{N^{1/d}} \}
$$
$Vol(S) = 2N\cdot \prod_{i=1}^{d} \frac{2}{N^{1/d}} = 2^{d+1}$ hence, you can apply Minkowski's Theorem, and find a point in S with integer coordinates. That point is your approximation.