Carry out the details of the following proof by contradiction that \(\sqrt{2}\) is irrational (this proof is due to R. Palais). If \(\sqrt{2}\) is rational, then \(n \sqrt{2}\) is a whole number for some whole number \(n\). Let \(n\) be the smallest such whole number and let \(m=n \sqrt{2}-n\).

(a) Prove that \(m
(b) Prove that \(m \sqrt{2}\) is a whole number.

Explain why (a) and (b) imply that \(\sqrt{2}\) is irrational.