Exercise 10, § 1

Let $a$ be a real number and $n$ an integer $> 0$, and consider the numbers $x_k = ka - [ka] \in [0, 1[ \quad (1 \leq k \leq n + 1)$. If $I_h$ denotes the interval

show that there exist two distinct indices $k, k'$ such that $x_k$ and $x_{k'}$ belong to the same interval $I_h$ (for a suitable value of $h$). Deduce that there exist two integers $p, q$ such that $1 \leq p \leq n$ and $|pa - q| \leq 1/n$.