Let $t_1, \ldots, t_n$ be strictly positive real numbers such that $\sum_{i=1}^n t_i = 1$ and let $N \geqslant n$ be an integer. We wish to show that there exist non-negative integers $p_1, \ldots, p_n$ and $q$ such that i) $1 \leqslant q \leqslant N^{n-1}$, ii) $\sum_{i=1}^n p_i = q$, iii) $\left|qt_1 - p_1\right| \leqslant \frac{n}{N}$, iv) for all $i = 2, \ldots, n$, $\left|qt_i - p_i\right| \leqslant \frac{1}{N}$. 16a. By considering the vectors with coordinates $\left(\{kt_2\}, \ldots, \{kt_n\}\right) \in [0,1[^{n-1}$ when $k$ ranges over $\{0, \ldots, N^{n-1}\}$, show that there exist integers $p_2, \ldots, p_n, q \geqslant 0$ satisfying conditions i) and iv). 16b. Conclude.
Let $t_1, \ldots, t_n$ be strictly positive real numbers such that $\sum_{i=1}^n t_i = 1$ and let $N \geqslant n$ be an integer. We wish to show that there exist non-negative integers $p_1, \ldots, p_n$ and $q$ such that
i) $1 \leqslant q \leqslant N^{n-1}$,
ii) $\sum_{i=1}^n p_i = q$,
iii) $\left|qt_1 - p_1\right| \leqslant \frac{n}{N}$,
iv) for all $i = 2, \ldots, n$, $\left|qt_i - p_i\right| \leqslant \frac{1}{N}$.
16a. By considering the vectors with coordinates $\left(\{kt_2\}, \ldots, \{kt_n\}\right) \in [0,1[^{n-1}$ when $k$ ranges over $\{0, \ldots, N^{n-1}\}$, show that there exist integers $p_2, \ldots, p_n, q \geqslant 0$ satisfying conditions i) and iv).
16b. Conclude.