Sum of Reciprocals in Base 10 with Zeroes Removed

Theorem
The infinite series


 * $\displaystyle \sum_{\map P n} \frac 1 n$

where $\map P n$ is the propositional function:
 * $\forall n \in \Z_{>0}: \map P n \iff$ the decimal representation of $n$ contains no instances of the digit $0$

converges to the approximate limit $23 \cdotp 10345 \ldots$

Proof
For each $k \in \N$, there are $9^k$ $k$-digit numbers containing no instances of the digit $0$.

Each of these numbers is at least $10^{k - 1}$.

Hence the reciprocals of each of these numbers is at most $\dfrac 1 {10^{k - 1}}$.

Thus:

showing that the sum converges.