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).