Let $N=\{n_1, n_2, ...\}$ Be a sequence of all positive integers whose decimal representation does not contain the digit $0$. Hence 45 is in $N$ but $10$ is not.
Show that $\sum _{k=1}^{\infty } \frac{1}{n_k}$ converges. Also show that the sum is less than $90$.
I've tried some of the usual "tests for convergence" like the ratio test but they come out inconclusive or they don't apply. What I was trying to work with was the harmonic series
$\sum _{k=1}^{\infty } \frac{1}{n}$
and then subtract out all the numbers that contain the digit zero but it wasn't working very well...