So this question has been asked before, see here, but instead of how to go from part 4 to part 5, I am having a difficult time proving part 4:
For each $\alpha > 0$ there exists a sequence of integers $\{n_1, n_2, \dots \}$, increasing in the weak sense, such that $p_{n_j} \sim \alpha j \qquad (j \to \infty)$.
How does it follow from the previous part?