The truncated harmonic series diverges.

The harmonic series diverges. Kempner has shown in 1914 that all terms containing the digit 9 can be removed without changing the divergence.
Here is a simple derivation: https://www.hs-augsburg.de/~mueckenh/HI/ p. 15.
I think that we can remove all terms containing 1, 2, 3, 4, 5, 6, 7, 8, 9, 0 in the denominator without changing the divergence.
Further I think we can remove every denominator containig any given number like 2025 without changing the divergence.
Further I think that we can remove the chain of all definable numbers without changing the divergence.
This is a proof of the huge set of dark numbers.
Regards, WM