Given a natural number k, let n be the quotient and r the remainder when k is divided by N. So k=nN+r, with 0≤r<N. Define F by . Then as k goes from nN to nN+(N-1), F(k) will go through all the values of f_n. Therefore the range of F (as k goes through all the natural numbers) will be the union of the sets E_n.