TITLE: Weak versions of Bertrand's postulate
QUESTION [6 upvotes]: 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.

REPLY [3 votes]: Pal Erdös' version can be found in "proofs from the book".  This was rather elementary.