Proof of irrationality of the Square root of 2

We will do a proof by contradiction:

Lets assume, that the square root of 2 is rational. Then you can write the square root of 2 as a fraction a over b. Lets assume that a and b are hole numbers and that you can not reduce a over b. Then you can do this:

It follows, that a is even.

Now we prove that b is even. For a we can write 2k.

It follows that b is even. That means, that a/2 and b/2 are hole numbers. It follows, that you can reduce a over b with 2. This is a contradiction, since we assumed that a over b can not be reduced.