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?
Consider $$n_j=\left[\frac{\alpha j}{\log j}\right].$$
Then $$p_{n_j}\sim n_j \log(n_j)\sim \frac{\alpha j}{\log j}\log\left(\frac{\alpha j}{\log j}\right)=\alpha j+\frac{\alpha j}{\log j}\log\left(\frac{\alpha}{\log j}\right)$$
$$=\alpha j+O\left(\frac{\alpha j\log \log j}{\log j}\right)\sim \alpha j.$$