Q:

Show that $ \sum_{n\in \mathbb {S}} \frac{1}{n} $ is convergent

Let $\mathbb {S} =\left \{ 1,2,3,...,9,11,12,...,19,21,...99,111,112,113... \right \} $
i.e, the positive integers set which contain zero digit is omitted.
Now show that $ \sum_{n\in \mathbb {S}}  \frac{1}{n} $ is convergent .
I really don't have no idea about how to prove this

A:

Note that \begin{eqnarray} \sum_{n\in \mathbb{S}\cap\{10^n, \cdots, 10^{n+1}-1\}}\frac{1}{n}<\frac{9^{n+1}}{10^n}\end{eqnarray} conclude that any partial sum is bounded and thus the series converges.

