Is it known that for every epsilon there is `N_0` such that all intervals of the form [N, (1+\epsilon)\*N], where N > `N_0`, contain prime numbers?



Strong Bertrand postulate
This follows from the Prime Number Theorem. Let π(n) be the number of primes less than n. Then π(n) ~ n/log(n); it follows π((1+ε)n)-π(n) -> ∞ as n -> ∞.


