Prove √2 is not a fraction or rational number

Prove that √2 is not a rational number.
Suppose that √2 really is a fraction, say x/y. We also assume that x/y is in its lowest terms.
Then √2=x/y or 2 = x²/y², cross multiply we have  x²=2y²
Now 2y² must be an even number, whatever y was. Hence x² must also be even. This means that x itself is even, as only the square of an even number is even. So x =2z, where z is some other number.
Then x²=2y²=4z², i.e y²=2z².
But 2z² is even, so therefore y² is also even. This means that y itself is even.
We have thus proved x and y are even. This cannot be true as x/y is in its lowest terms.  Our original assumption that √2 = x/y has led to a contradiction.
We must then conclude that √2 is not a fraction or a rational number.