2001 IMO Shortlist S03

Let $x_1,x_2,\ldots,x_n$ be arbitrary real numbers. Prove the inequality

$$

\frac{x_1}{1+x_1^2} + \frac{x_2}{1+x_1^2 + x_2^2} + \cdots +

\frac{x_n}{1 + x_1^2 + \cdots + x_n^2} < \sqrt{n}.

$$

令 $x_1,x_2,\ldots,x_n$ 为任意实数。证明不等式

$$\frac{x_1}{1+x_1^2} + \frac{x_2}{1+x_1^2 + x_2^2} + \cdots +$$
\frac{x_n}{1 + x_1^2 + \cdots + x_n^2} < \sqrt{n}。