I was wondering if someone could explain the intuition behind what appears to be a straightforward consequence of the Prime Number Theorem. Let $(x_k)_{k\geq 1}$ be a nondecreasing sequence in $ \mathbb{R}_{>0}$ such that $|\{k \in \mathbb{N} : x_k \leq x\}| \sim \alpha \frac{x}{\log x}$, where $\alpha > 0$ is a constant. The Prime Number Theorem then obviously implies that $|\{k \in \mathbb{N} : x_k \leq x\}| \sim \alpha \pi (x)$, where $\pi(x)$ is the standard notation for the number of primes less than or equal to $x$. It needs to be proved that $x_n \sim \frac{1}{\alpha}n \log n$, and I have already completed the proof that $p_n \sim n \log n$, where $p_n$ denotes the $n$th prime, so the problem reduces to showing that $\pi(x_n) \sim \frac{1}{\alpha} \pi(p_n)$ implies $x_n \sim \frac{1}{\alpha}p_n$.
I have reasoned thus far that $\alpha \frac{x_n}{\log x_n} \sim |\{k \in \mathbb{N} : x_k \leq x_n\}| = n = \pi(p_n) \sim \frac{p_n}{\log p_n}$, but I am struggling to fill in the final details leading to the conclusion $x_n \sim \frac{1}{\alpha}p_n$, most likely because I am lacking intuition about what an asymptotic relation between two different values of $\pi(\cdot)$ tells us about the asymptotic relation between its arguments. Thank you in advance for your assistance!