Proposition 2.3.6

Propostion 2.3.6 (Multiplication preserves order). If \(a,b\) are natural numbers such that \(a<b\), and \(c\) is positive, then \(ac<bc\).

\(Proof\). Since \(a<b\), we have \(b=a+d\) for some positive \(d\). Multiplying by \(c\) and using the distributive law we obtain \(bc=ac+dc\). Since \(d\) is positive, and \(c\) is positive, \(dc\) is positive, and hence \(ac<bc\) as desired. \(\square\)

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Corollary 2.3.7

\(Proof\). By the trichotomy of order (Proposition 2.2.1), we have three cases: \(a<b,a=b,a>b\). Suppose first that \(a<b\), then by Proposition 2.3.6 we have \(ac<bc\), a contradiction. We can obtain a similar contradiction when \(a>b\). Thus the only possibility is that \(a=b\), as desired. \(\square\)
Chapter2

Proof by Tao.