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.