Why is $\lim_{n \to \infty} \frac{1}{n} \zeta_\mathbb{N} (A\cap[1,n])$ not defined for numbers whose decimal representation starts with $3$?

Question

See the comments here: $A \subseteq \mathbb{N}$ for which $\mu(A)=\lim_{n \to\infty}\frac{1}{n} \zeta \big|_\mathbb{N} (A \cap[1,n])$ is not defined

($\zeta_\mathbb{N}$ describes the counting measure on $\mathbb{N}$)

What does he mean with $[1,2999999]$ and $[1,3999999]$ and with density?

Answer 1

Fewer than $1/10$ of the first $2999999$ numbers begin with a $3$. More than $1/4$ of the first $3999999$ numbers begin with a $3$. So the fraction doesn't settle down to a single limit. It continues to vary by more than $0.15$ forever.