Euclid's Proof that 2^.5 is Irrational

assume the contrary:
2^.5 = p/q
square both lines.
2 = p²/q²
multiply both sides by q²
2q² = p²
p² must be even, for it is equal twice the value of an integer. ( q² ) p is therefore even as well, for if the square of a number is even, the number is as well.
because p is even, it can be written as 2m, where m is some other whole number. placed into the equation, it reads:
2q² = (2m)² = 4m²
divide both sides by 2.
q² = 2m²
with the same arguments used previously, q must also be even, and may be represented as 2n. same steps, resulting in
2^.5 = p/q = 2m/2n
lose the 2s.
2^.5 = m/n
This process can continue indefinitely, meaning the fraction is infinitely reducable. A quality of a fraction is that it may be represented as a ratio between two relatively prime integers. (like 10/20 can be reduced to 5/10, and finally to 1/2)
yeah... QED