elementary number theory - Proving that the square root of 5 is irrational

Prove that $\sqrt{5}$ is irrational.

I begin with the identity $(\sqrt{5} + 2 )(\sqrt{5} - 2 ) = 1$.

Then I am told to extract $\sqrt{5}$ from the first or second factor and consider it to be $\frac{m}{n}$ so I should replace it in both sides.

I have

$$\frac{m}{n} = (\frac{1}{\frac{m}{n}} + 2) + 2.$$

I am also told to work on the right side until I have a denominator less than $n$ and I have to explain the reasoning.

Then I have to prove this is false by contradiction.

Right now my main problem is I can't get a denominator less than $n$.