The theorem says that for all n≥3 there is a prime number between n and 2n. This proof was published by Paul Erdos in 1932, when he was 19.
It includes a lot of steps, but each step needs no more than A-level maths (other than the notation Π for a product of a sequence, what you get by multiplying together all the numbers in it, in the same way that Σ is used for what you get by adding together all the numbers in a sequence).
The idea that for all n≥3 there is a prime number between n and 2n had been put forward by Joseph Bertrand almost 100 years earlier. He checked the idea for all n up to 3 million, but couldn’t prove it.
Pafnuty Chebyshev proved the theorem a long time before Erdos, but by more complicated methods.