Bertrand’s postulate

We have seen that the sequence of prime numbers 2, 3, 5, 7, … is infinite. To see that the size of its gaps is not bounded, let N := 2 • 3 • 5•...• p denote the product of all prime numbers that are smaller than k + 2, and note that none of the k numbers $$N - 2,N + 3,N + 4,...,N + k,N + (k + 1)$$ is prime, since for 2≤ i ≤ k + 1 we know that i has a prime factor that is smaller than k + 2, and this factor also divides N, and hence also N + i. With this recipe, we find, for example, for k = 10 that none of the ten numbers $$2312,2313,2314,...,2321$$ is prime.