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).

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    $\begingroup$ You can prove by elementary means $p_n=O(n\log n)$. $\endgroup$ – karvens Oct 21 '14 at 5:09
  • $\begingroup$ The statement is not true for $p_1=2$, so you need an exception for that case. $\endgroup$ – Esteban Crespi Oct 21 '14 at 5:50
  • $\begingroup$ @karvens: But how does it conclude that $p_n<n^2$, can you elaborate? $\endgroup$ – user 170039 Oct 21 '14 at 6:10
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    $\begingroup$ If $p_n=O(n\log n)$, we have $p_n<Cn\log n<n^2$ for all sufficiently large $n$. $\endgroup$ – karvens Oct 21 '14 at 6:14
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    $\begingroup$ @user170039 Chebyshev already proved something of the strength $p_n \le (1.2 + o(1)) n \log n$; this was nearly 50 years before PNT was established. The proof that karvens refers to is probably the simpler one due to Erdős. $\endgroup$ – Erick Wong Oct 21 '14 at 8:52

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.


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