I have been working through the exercises in Tenenbaum's "Introduction to analytic and probabilistic number theory" book, and I am stumped here (Exercise I.1.6). This is not a homework assignment, but just rather for my own edification. I know Tenenbaum's exercises are usually considered quite hard, so I don't feel as embarrassed to ask this question!
For those not aware (I suspect most experts know this exercise and book inside and out), the homework problem reads as follows:
Set $d_n = p_{n+1} - p_n$.
Show that $p_n \sim n \log n \qquad (n \to \infty)$
$\sum_{1 < n \leq x} d_n / \log x \sim x \qquad (x \to \infty)$
$\liminf_{n \to \infty} \frac{d_n}{\log n} \leq 1 \leq \limsup_{n \to \infty} \frac{d_n}{\log n}$
For each $\alpha > 0$ there exists a sequence of integers $\{n_1, n_2, \dots \}$, increasing in the weak sense, such that $p_{n_j} \sim \alpha j \qquad (j \to \infty)$.
The set of rational numbers of the form p/p', where $p$ and p' are prime is dense in $[0, \infty)$.
My questions are two-fold:
How does part 5 of this exercise follow from part 4?
Are there different proofs (elementary or not) of question 5 that do not follows this route?