Question 61
Let $x_{1}, x_{2}, \ldots, x_{n}$ be positive integers. Suppose that, in their decimal representation, none of the $x_{i}$ is an "extension" of another $x_{j}$. For example, $123$ is an extension of $12$, and $459$ is an extension of $4$, but $134$ is not an extension of $123$.
Prove that
$$
\frac{1}{x_{1}}+\cdots+\frac{1}{x_{n}}\lt 3
$$
Prove that
$$
\frac{1}{x_{1}}+\cdots+\frac{1}{x_{n}}\lt 3
$$