Erdős proved that for any positive integer k, there is a natural number N such that for all n > N, there are at least k primes between n and 2n. An equivalent statement had been proved earlier by Ramanujan (see Ramanujan prime).

Erdős [1] 对 Bertrand 假设给出了一个非常漂亮的初等证明.

Erdős 证明了对任意正整数 $k$, 存在 $N$, 使得对所有 $n>N$, 至少有 $k$ 个素数介于 $n$ 和 $2n$ 之间.

http://en.wikipedia.org/wiki/Bertrand\'s_postulate

[1] P. Erdős, Beweis eines Satzes von Tschebyschef, Acta Sci. Math. (Szeged) 5 (1930–1932), 194–198.

[2] David Galvin, Erd˝os\'s proof of Bertrand\'s postulate. [PDF]