The Prime Number Theorem.
Chebyshev proved that if $$\lim_{n\to\infty}{\pi(n)\log n\over n}$$ exists (here, $\pi(n)$ is the number of primes up to $n$), then it equals $1$. Fifty years passed after that before Hadamard and de la Vallée Poussin (independently) proved that the limit exists.
