Problem 4

Prove that for any positive integer

k

, there exists an arithmetic sequence

\frac{a_1}{b_1},\frac{a_2}{b_2},\ldots,\frac{a_k}{b_k}

of rational numbers, where

a_i,b_i

are relatively prime positive integers for each

i=1,2,\ldots,k

, such that the positive integers

a_1,b_1,a_2,b_2,\ldots,a_k,b_k

are all distinct.

Step 4 of 5: Separate a numerator from a denominator

a_i\ge\frac{K^2}{k}+1=K+1> K\ge b_j

Detailed analysis

For every i, since i≤k, a_i≥K^2/k+1=K+1. Every denominator b_j=K/j is at most K. Thus no numerator equals any denominator.