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  • https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/An_Introduction_to_Number_Theory_(Veerman)/01%3A_A_Quick_Tour_of_Number_Theory/1.02%3A_Rational_and_Irrational_Numbers
    Note that any non-empty set \(S\) of integers with a lower bound can be transformed by addition of a integer \(b \in N_{0}\) into a non-empty \(S+b\) in \(N_{0}\). The advantage of expressing a ration...Note that any non-empty set \(S\) of integers with a lower bound can be transformed by addition of a integer \(b \in N_{0}\) into a non-empty \(S+b\) in \(N_{0}\). The advantage of expressing a rational number as the solution of a degree 1 polynomial, however, is that it naturally leads to Definition 1.12. The crux of the following proof is that we take an interval and scale it up until we know there is an integer in it, and then scale it back down.

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