Write each of these five numbers as $\tan\theta_k$ where $k=1$, $2$, $3$, $4$, $5$, and $0 \le \theta_k \le 180^\circ$ Dividing the range of $[0, 180^\circ]$ into four sub-divisions: $[0, 45]$, $[45^\circ, 90^\circ]$, $[90^\circ, 135^\circ]$, and $[135^\circ, 180^\circ]$. By the pigeonhole principle (The pigeonhole principle is discussed in the book  Art of Thinking ), at least two of them must fall into one sub vision. Let them be $\theta_i$ and $\theta_j$. Then $$\Big{|}\frac{x-y}{1+xy}\Big|=\Big|\frac{\tan\theta_i - \tan\theta_j}{1+\theta_i \theta_j}\Big|=|\tan(\theta_i-\theta_j)|\le|\tan 45^\circ| = 1$$ $$\therefore\quad |x-y| < |1+xy|$$