Proof of $p_n<n^2$ by Elementary Means

Question

Is there any proof of the inequality $p_n<n^2$ (for all sufficiently large $n$) by elementary means and without using Prime Number Theorem?

I searched in google but in vain. The results that I have found (and from which the inequality can be proved) are mostly results that arises as a consequence of PNT.

So, is there any such proof? If so, then please include a link of the paper in your answer (or comment).

Answer 1

As said in the comments, Chebyshev already proved by elementary means that $p_n<C n \log(n)$ for all $n\ge n_0$. As there was explicitly asked for a reference, I include one (of many): Theorem $5.1$ here.