# Proof that sequence of factorials is a Benford sequence without Weyl criterion

I know of a short proof that shows that the proportion of powers of $$2$$ with first digit '$$1$$' is $$\log_{10}(2)$$. It involes analysing the intervals of positive numbers that start with $$1$$: $$[1,1), [10, 19), [100,199],...$$

Now I'm trying to construct a similar proof for the sequence of factorials $$(n!)$$ that does not use the Weyl criterion.

Is there some short proof, or if not, do you know any other sequence besides the powers of some number that is Benford and it is easy to proof this?

• If the Weyl-criterion works (which you could work out in the case it does) , why do you reject to use it ? Apr 24 at 13:42
• I'm working on a thesis and the Weyl criterion will be introduced later. For the introduction I'd like to present some sequences and show that the proportions of significant digits follow some pattern. I've found a short proof about the powers of $2$ and it was very intuitive so I wondered if there are other easy proofs for Benford sequences. Apr 24 at 13:49