Question 29
Let $a_{1}\lt a_{2}\lt \ldots\lt a_{n}$ be a sequence of positive integers with the property that for $i\lt j$, the decimal representation of $a_{j}$ does not begin with that of $a_{i}$ (for example, the numbers 137 and 13729 cannot both appear in the sequence). Prove that is true
$$
\sum_{i=1}^{n} \frac{1}{a_{i}} \leq \frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{9}
$$
$$
\sum_{i=1}^{n} \frac{1}{a_{i}} \leq \frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{9}
$$