The existence of dark numbers proven by the thinned out harmonic series

The harmonic series diverges. Kempner has shown in 1914 that when all terms containing the digit 9 are removed, the series converges.
That means that the terms containing 9 diverge. Same is true when all terms containing 8 are removed. That means all terms containing 8 and 9 simultaneously diverge.
We can continue and remove all terms containing 1, 2, 3, 4, 5, 6, 7, 8, 9, 0  in the denominator without changing this. That means that only the terms containing all these digits together constitute the diverging series. (*)
But that's not the end! We can remove any number, like 2025, and the remaining series will converge. For proof use base 2026. This extends to every definable number. Therefore the diverging part of the harmonic series is constituted only by terms containing a digit sequence of all definable numbers.
Note that here not only the first terms are cut off but that many following terms are excluded from the diverging remainder.
This is a proof of the huge set of undefinable or dark numbers.
(*) At this point the diverging series starts with the smallest term 1023456789 and contains further terms like 1203456789 or 1234567891010 or 123456789111 or 1234567891011. Only those containing the digit sequence 10 will survive the next step, and only those containing the digit sequence 1234567891011 (where the order of the first nine digits is irrelevant) will survive the next step.
Regards, WM