prime number theorem
The number of prime numbers less than x is about x/log(x). Here "is about" means that the ratio of the two things tends to 1 as x tends to infinity. This was first conjectured by Gauss in the early 19th century, and was proved (independently) by Hadamard and de la Vall'ee Poussin in 1896. Their proofs relied on complex analysis, but Erdös and Selberg later found an "elementary" proof.