TITLE: Existence of the natural density of the strictly-increasing sequence of positive integer?
QUESTION [6 upvotes]: Let $A=\{a_n\}$ is a strictly-increasing sequence of positive integer. The natural density of this sequence is defined by $\delta(A)=\lim_{n\rightarrow \infty} \frac{A(n)}{n}$ whenever the limit exists and where $A(n)$ is the number of elements of $A$ not exceeding $n$. Is there a strictly-increasing sequence of positive integer $A=\{a_n\}$ such that $\delta(A)$ does not exists?

REPLY [15 votes]: Consider the sequence of integers with an odd number of decimal digits.

REPLY [9 votes]: Yes. Start with $1$. Then omit enough integers to reduce the ratio below $\frac{1}2$. Then include enough consecutive integers to increase the ratio above $1-\frac{1}3$. Then omit enough to reduce it below $\frac{1}4$. Then include enough to increase it above $1-\frac{1}5$. Keep going.