A Shorter Proof of Rosser's Theorem Without Using The Prime Number Theorem

While researching on the elementary proof of Bertrand's Postulate I came to know about a theorem of Rosser's which states that $p_n$ $>$ $n$ $\text{ln}$ $n$. I have seen Rosser's original proof and I am interested in shorter proof of the result. It will be best if it could be proved using elementary methods. Any suggestion will be appreciated.

There is an elementary proof by Chebyshev, which shows that $p_n>\alpha n\log(n)$ for a constant $a<1$ close to $1$. However, for $\alpha=1$ one needs more, i.e., one needs the techniques of Rosser and Schoenfeld, using zero-free regions for the Riemann zeta function. I think there is no easier proof than the one of Rosser. Piere Dusart has sharpened the estimates on the $n$-th prime. This is perhaps easier to read. References can be found in the answers here.