Recently I have been reading this post and I have noted something significant in the fake argument. As one can easily see that the basic idea behind the argument had been to show that the sequence $x_n=\dfrac{\pi(n)\ln n}{n}$ is strictly decreasing for all $n$.
But, notice that the Prime Number Theorem is equivalent to the statement that $\displaystyle\lim_{n \to \infty}\dfrac{n\ln p_n}{p_n}=1$. So, the argument of the post previously mentioned can be used to actually prove the Prime Number Theorem if it can be shown that $n>\dfrac{p_n}{\ln p_n}$ for all sufficiently large $n$.
So, is there any elementary method to prove this?
Edit:- Yesterday while searching the internet, I have found that this paper gives a proof of the result (see page 81). But since I am not an expert in this field, I am a bit skeptic about the method of proof. Is there any subtle assumption in the argument that uses PNT?