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Proof

(related to Corollary: Existence of Natural Numbers Exceeding Positive Real Numbers (Archimedian Principle))

• By hypothesis, the set of real numbers $\mathbb R$ is ordered".
• Since $1 > 0$ holds in any ordered field, we have \(1 > 0\) also in $\mathbb R$.
• Thus $1$ is a positive real number.
• The Archimedian axiom ensures the existence of a natural number \(n\) such that \(nx > y\) for any positive real numbers \(x,y\), thus also for the positive numbers $1,y.$
• It follows that for every positive real number \(y > 0\) there exists a natural number \(n\) with \(y < n\).
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References

Bibliography

1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983
2. Kramer Jürg, von Pippich, Anna-Maria: "Von den natürlichen Zahlen zu den Quaternionen", Springer-Spektrum, 2013