 Chebyshev's papers predated Riemann's celebrated memoir of 1859, and he succeeded in proving a slightly weaker form of the asymptotic law, namely, that if the limit as goes to infinity of exists at all, then it is necessarily equal to one.He was able to prove unconditionally that this ratio is bounded above and below by two explicitly given constants near 1, for all sufficiently large .Although Chebyshev's paper did not prove the Prime Number Theorem, his estimates for were strong enough for him to prove Bertrand's postulate that there exists a prime number between and for any integer .