91Ó°ÊÓ

In this proof, let ℤ be the set of integers and ℚ be the set of rational numbers. The additive group of rational numbers is denoted by (ℚ, +), and the additive group of integers is denoted by (ℤ, +). The goal is to show that the index of the subgroup (ℤ, +) in the group (ℚ, +) is infinite. This means that there are infinitely many distinct cosets of (ℤ, +) in (ℚ, +).

To show that there are infinitely many distinct cosets of (ℤ, +) in (ℚ, +), consider the set of cosets given by the set {n + 1/n : n ∈ ℕ}, where ℕ is the set of natural numbers. For each natural number n, we have a coset n + 1/n + ℤ, which represents all the rational numbers obtained by adding the elements of ℤ to n + 1/n. We claim that each coset n + 1/n + ℤ is distinct. To prove this, suppose we have two cosets m + 1/m + ℤ and n + 1/n + ℤ with m ≠ n, and assume that they are NOT distinct. In this case, there would exist integers i and j such that m + 1/m = n + 1/n + i - j. Rearranging the equation, we get (m - n) + (1/m - 1/n) = i - j. Since (m - n) is an integer, to satisfy this equation, we must have that (1/m - 1/n) is also an integer. However, we know that 1/m - 1/n ≠ 0 for m ≠ n, and both (1/m, 1/n) are rational numbers, so their difference must be a rational number and not an integer. This contradicts our assumption, and thus, m + 1/m + ℤ and n + 1/n + ℤ are distinct cosets.

Now we have a set of distinct cosets {n + 1/n + ℤ : n ∈ ℕ}. Since ℕ is countably infinite, there are infinitely many natural numbers, and hence, infinitely many distinct cosets in this set. This shows that the index of the subgroup (ℤ, +) in the additive group of rational numbers (ℚ, +) is infinite, as required.