BALTIC-WAY 2000 Problem 12

Let $x_{1}, x_{2}, \ldots, x_{n}$ be positive integers such that no one of them is an initial fragment of any other (for example, 12 is an initial fragment of $\underline{12}, \underline{12} 5$ and

12405). Prove that

$$\frac{1}{x_{1}}+\frac{1}{x_{2}}+\cdots+\frac{1}{x_{n}}<3$$