Let $1,2,3,4,5,6,7,8,9,11,12,\ldots$ be the sequence of all positive integers which do not contain the digit zero. Write $\{a_n\}$ for this sequence. By comparing with a geometric series, show that $\sum\limits_{k=1}^n \dfrac{1}{a_k} < 90$.
My attempts:
I could not attempt the first part of the question to find a formula for generalising $\{a_n\}$ but for the second part, I tried,
$$\sum_n \dfrac{1}{a_n}<1+\dfrac12+\dfrac12+\dfrac14+\dfrac14+\dfrac14+\dfrac14+\ldots$$
but could not get anything worth of mention. Please help.