Exercise 9, § 1

\* (Minkowski's Theorem) If a, b, c, d are real numbers, and $D = \begin{vmatrix} a & b \\ c & d \end{vmatrix}$, then the inequalities $|na + mb| \leq A, |nc + md| \leq B$ have nonzero integer solutions $(n, m)$ if $A > 0, B > 0, D > 0$ and $AB \geq D$, and such solutions are finite in number (consider the discrete subgroup G of $\mathbf{R}^2$ generated by $(a, c)$ and $(b, d)$, and show that the rectangle E defined by the relations $0 \leq x \leq A, 0 \leq y \leq B$, whose area is greater than that of the parallelogram constructed on the vectors $(a, c)$ and $(b, d)$, contains two distinct vectors which are congruent mod G; argue by contradiction, noticing that otherwise the $p^2$ rectangles obtained from E by means of the translations $(ha + kb, hc + kd)$, where $0 \leq h < p$ and $0 \leq k < p$, would be pairwise disjoint).