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that \(\sqrt{2}\) is irrational. Suppose that \(\sqrt{2}\in\mathbb Q\). So there exists an integer \(k>0\) such that \(k\sqrt{2}\in\mathbb Z\). Let \(n\) be the smallest one with this property and take \(m=n(\sqrt{2}-1)\). We observe that \(m\) has the following properties:
\[
1. \ m∈Z \\
2. \ m>0 \\
3. \ m\sqrt{2}\in\mathbb Z \\
4. \ m<n
\]
and so we come to a contradiction! Q.E.D.

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