Existence of pair of primes with difference $2n$ for every integer $n$

Question

So this might be a silly question. There is the open problem of proving there are infinitely many pairs of primes with a fixed even difference, for any given even number. This maybe implies that it's known that given any even number, there exists a pair of primes with that difference. Is this a hard thing to prove if it is even known, or is it a simple fact?