Weak versions of Bertrand's postulate We are interested in the following statement: 
For each $n>1$ and $x>2$ there is at least one prime $p$ satisfying $x<p<n x$.
For $n=2$ we get precisely the Bertrand's postulate which is true. As corollaries, the statements for arbitrary $n\geqslant 2$ are true. However, I am interested maybe there exist independent proofs (prossibly short and elementary) for some of the cases $n>2$.
Thanks for your help.  

The proof of Bertrand's postulate given in e.g. Proofs from The BOOK isn't too involved... it is essentially as given by Wikipedia.
