I have started reading Gelford & Linnik's elementary methods in analytic number theory (1965).
They define a sequence $A$ of integers as:
$$0, a_1, a_2,a_3,\dots$$
where
$$0 < a_1 < a_2 < a_3 < \dots$$
Let:
$$A(n) = \sum_{0<{a_i} \le n} 1$$
So that:
$$0\le\frac{A(n)}{n}\le1$$
I am following their explanations up to this point. Then, the following definition of density $d(A)$ is offered:
$$d(A) = \inf_n \frac{A(n)}{n}$$
At this point, I am not clear on how the definition maps to the examples. I found a wikipedia article on Schnirelmann density but that didn't help. I'll reread it this evening.
Gelford & Linnik provide examples of density. I would greatly appreciate it if someone could explain me how the definition above maps to these examples.
Here are three examples from the section:
(1) If $1 \notin A$, then $d(A) = 0$
(2) $d(A) = 1$ if and only if $A$ contains all the positive integers.
(3) The densities of the sequences of squares, cubes, and prime numbers equal $0$.
(1) Clearly if $1\notin A$ then $A(1)=0$. (we try to find the (trivial) number of natural numbers not exceeding $1$ that are not $1$)
But generally, $d(A)\geq 0$ and $\frac{A(1)}{1}=0$
So, inf $\frac{A(n)}{n}=0$.
(2) Suppose that there exists a positive integer $k$ which is the least one that is not contained in $A$. Then $\frac{A(k)}{k}<1$ and so, (clearly) inf $\frac{A(n)}{n}\leq A(k)<1$
(3)you can see that the squares,cubes etc are getting more and more rare among the positive integers so the function whose infimum you want to calculate is strictly decreasing.