How to proof square root of 2 is irrational number

Greek ancient find square root of 2 wasn’t rational although square root of 2 was the length of hypotenuse with upright side triangles was 1. The number cannot write as result from integer number. So the square root of 2 is irrational.
The proof:
Rational number is the number which can be stated in ratio a over b, a and b the integer number which do not have factor partner and b unequal 0
If square root of 2 is rational number, then :
Square root of 2 equals a over b times r
¬2 equals a square over b square
¬a square equals 2 times b square (2b square is the integer number, integer times 2 is even number )
¬a square equals even
¬a equals even………(1)
¬a equals 2n (n is integer number)
¬a square equals 4n square
¬2 times b square equals 4n square
¬b square equals 2n square
¬b square is even………(2)
¬b equals even

From equation 1 and 2 we know that a and b are even.
a and b are even, so they have factor partner that is 2. The entire step is right, that is the opposite with the definition of the rational number. It means the proof is wrong, so square root of 2 is irrational.