Real numbers/Ordering axioms/Archimedean/Implications/Fact/Proof2

Proof

(1). We consider ${}y/x$. Because of the Archimedean axiom there exists some natural number ${}n$ with ${}n\geq y/x$. Since ${}x$ is positive, due to fact  (6) also ${}nx\geq y$ holds. For (2) and (3) see exercise.

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