There are 3 standard proofs of Bertrand's Postulate:


(1) Chebyshev's original proof


(2) Ramanujan's simplification of Chebyshev's proof


(3) Erdos's proof


I recently learned about the very simple proof that if the Goldbach conjecture is true, then Bertrand's postulate follows (see [here](http://www.proofwiki.org/wiki/Goldbach_implies_Bertrand)).


Does anyone know of any other proofs? There are recent proofs that extend Bertrand's postulate to show that there is always a prime in $2n$/$3n$ and $3n$/$4n$. 


I am wondering if there aren't other lesser known proofs that take a different approach to establish the existence of a prime between $n$ and $2n$.


Thanks,


-Larry



Are there any interesting or lesser known proofs related to Bertrand's Postulate
Bertrand's Postulate follows as a direct consequence of the following theorem of J.J. Sylvester: 


**Theorem (Sylvester, 1892):** Let $k$ be a positive integer. Then at least one of any $k$ consecutive integers greater than $k$ is divisible by a prime greater than $k$.


(For comparison: Chebyshev's analytic proof dates to 1850; Erdos' elementary proof dates to 1932.)


See Theorem 6 (p. 6) in <http://www.math.sc.edu/~filaseta/papers/schurpaper.pdf>, from which I quote: 


"The theorem implies immediately that for any positive integer $k$, one of $k+1, k+2, \ldots, 2k$ is a prime (since one of these integers must be divisible by a prime $\geq k+1).$"


A copy of the Sylvester paper can be found [here](http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN599484047_0021&DMDID=DMDLOG_0003&IDDOC=644640).




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**Edit:** An article in the AMM (Aug/Sept, 2013) presents a revised version of Ramanujan's proof of Bertrand's Postulate; in particular, in which the use of Stirling's formula is eliminated. The citation is:


*Ramanujan’s Proof of Bertrand’s Postulate.* Jaban Meher, M. Ram Murty. The American Mathematical Monthly, Vol. 120, No. 7 (August–September 2013), pp. 650-653. <http://www.jstor.org/stable/10.4169/amer.math.monthly.120.07.650>.


