The proof isn't referring to n, n+1, and n+2 -- it refers to n, n+1, and n(n+1) + 1. In your example, this would be 2, 3, and 2*3 + 1 = 7, which in fact include

The proof isn't referring to n, n+1, and n+2 -- it refers to n, n+1, and n(n+1) + 1. In your example, this would be 2, 3, and 2*3 + 1 = 7, which in fact include 3 prime factors.