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 5 of 5: Conclude distinctness

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\left\{a_1,b_1,\ldots,a_k,b_k\right\}\text{ has }2k\text{ distinct positive integers}

Detailed analysis

The numerators are pairwise distinct, the denominators are pairwise distinct, and the two families are disjoint. Therefore all 2k positive integers a_1,b_1,…,a_k,b_k are distinct, and the constructed sequence proves the claim.