If $\{n_k\}$ is the set of natural numbers with no 0 in their decimal expansion, $\sum_{k=1}^\infty \frac{1}{n_k}$ converges to a number less than 90

Question

Let ${\{n_1,n_2,…\} }$ be the set of natural numbers that do not use the digit 0 in their decimal expansion. Then, the series $\sum_{k=1}^\infty \frac{1}{n_k}$ converges to a number less than 90.

Is it ok to consider the series as $$\sum_{k=1}^\infty \frac{1}{n} - \sum_{i=1}^\infty \frac{1}{10^i}$$

This is an exercise problem in "Mathematical analysis" by ${Tom.M.Apostol.}$ I didn't understand the proof given for a similar question in the site already. I'm not good at permutations and combinations and inclusion and exclusion. Can anyone help with the proof with elementary calculations.