Let $A, B$ be subsets of the natural numbers with $A \subseteq B$. The asymptotic density of $A$ in $B$ is $\lim_{N\to \infty}\frac{\text{number of elements of }A\text{ below }N}{\text{number of elements of }B\text{ below }N}.$ The Dirichlet density is
$\lim_{s\to 1^+}\frac{\sum\limits_{n\in A}n^{-s}}{\sum\limits_{n\in B}n^{-s}},$
with $s$ going to $1$ from the right, so the sums are finite. I want to show that if the natural density exists, then the Dirichlet density also exists and is the same. A paper by Bell and Burris proves this in a more general context, but it's a bit involved. Lang says this is "an easy exercise" so I must be missing something. Can someone help me out?
Thanks.
(Edit: I forgot to say that we also require $\sum_{n\in B} n^{-1} = \infty$, otherwise there are obvious counterexamples.)