Square Root of 5 is Irrational

Suppose  
\[\sqrt{5}\]
  is rational. Let  
\[\sqrt{5}= \frac{m}{n}\]
, where this fraction cannot be simplified.. Then  
\[\sqrt{5}= \frac{ \sqrt{5}( \sqrt{5} -2)}{\sqrt{5}-2}= \frac{5-2 \sqrt{5}}{\sqrt{5}-2}=\frac{5-2m/n}{m/n-2}=\frac{5m-2n}{m-2n}\]

Then  
\[m-2n=( \sqrt{5}-2)n\]
  and  
\[0 \lt \sqrt{5}=2 \lt 1 \]
  so  
\[0 \lt m-2n \lt n\]
. The assumption that  
\[\sqrt{5}\]
  is rational leads to s simpler rational fraction, but we assumed that  
\[\frac{m}{n}\]
  was already as simple as possible - a contradiction, so  
\[\sqrt{5}\]
  is irrational.

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