IMO Shortlist 2001 problem A3

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

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

Izvor: Međunarodna matematička olimpijada, shortlist 2001

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IMO Shortlist 2001 problem A3