This question is about natural density $d(A) = \lim_{x\to\infty}\frac{1}{x}\sharp\{n \leq x: n \in A\}$. I'm trying to prove that when either that limit or this limit:
$\lim_{n\to \infty} \frac{n}{a_n}$ exists, then the other exists and they're equal.
I've got, working backwards, that I'm trying to prove $\forall \epsilon \gt 0, \exists n_{\epsilon} : \forall n \geq n_{\epsilon}, |\frac{n}{a_n} - d(A)| \lt \epsilon$. But I'm not seeing what to do next.
($a_n$ is the ordered sequence that is $A$: $a_1 \lt a_2 \lt \dots$. In other words take the set $A$ and retrieve its elements in order to generate the sequence.)