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Fermat's Method of Infinite Descent

Fermat's 'method of infinite descent' is used to prove the irrationality of square roots.
Suppose for example that  
\[\sqrt{2}\]
  is rational so that  
\[\sqrt{2} = \frac{a}{b}\]
  and cannot be simplified
Then  
\[2= \frac{a^2}{b^2} \rightarrow 2b^2 =a^2\]

This means that  
\[a\]
  is even so we can write  
\[a=2x\]

Hence  
\[2b^2 =(2x)^2=4x^2\]

We can cancel by 2 to give  
\[b^2=2x^2 \rightarrow 2=\frac{b^2}{z^2} \rightarrow \sqrt{2} = \frac{b}{x}\]

This contradicts the initial assumption that  
\[\frac{a}{b}\]
  cannot be simplified, hence  
\[\sqrt{2}\]
  is irrational.

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