Let $(a_n)$ be a strictly increasing sequence, and let $P(x) := \# \{a_n \mid a_n \le x \}$ be the corresponding counting function.
We are given that
$$ P(x) \sim \frac{x}{\log x} $$
from which we want to deduce that
$$ a_n \sim n \log n $$
I am a bit stuck as to how to progress with this. I have tried starting with the fact that $P(a_n) = n$ which gives us that
$$ \frac{a_n}{\log a_n} \sim n $$
but since $\frac{x}{\log x}$ is not easily invertible (i.e. we would have to use something like the Lambert W function), this is not as straightforward as hoped.
Working backwards, it is easy to show that, if we denote $b_n := n \log n$, then
$$ \frac{b_n}{\log b_n} \sim n \sim \frac{a_n}{\log a_n} $$
But can we deduce from this that $a_n \sim b_n$? (For what it's worth, I know that the converse of this is true.)