The set of rational numbers of the form p/p', where p and p' are prime, is dense in $[0, \infty)$

From question 4 you can easily show that for any $\alpha > 0$, there exists a sequence $(q_n, q'_n)$ of prime numbers such that $q_n/q'_n$ converges to $\alpha$.

You just have to take the sequence $\{n_1, n_2, \dots\}$ and use the subsequence of the terms with prime index. So you can build this sequence $\{(p_1, p_{n_{p_1}}), (p_2, p_{n_{p_2}}), \dots \}$. Each pair in this new sequence in composed of prime numbers, and from the fact that $p_{n_j} / j \rightarrow \alpha$ you get that $p_{n_{p_i}}/p_i \rightarrow \alpha$ as $i \rightarrow \infty$.

Therefore $\alpha$ is in the closure of the rational numbers that are quotient of primes. Since it is true for any $\alpha > 0$, this set is dense in $[0, \infty)$.