Show that for every positive integer N there is an even number K so that there are more than N pairs of successive primes such that K is the difference between these successive primes (Hint: Use prime number theorem)

So, this is how far I've gotten..

Consider a list of primes and consider the difference of the successive primes

p_1,p_2,...,p_t

p_t - p_(t-1) = K

p_(t-1) - p_(t-2) = K

.

.

.

p_2 - p_1 = K

From here, I've been told to consider the sum of these differences and I should arrive to a conclusion that should contradict the prime number theorem.