Assume √2 is rational, so
√2 = p/q
for some relatively prime integers p,q - meaning we can't reduce p/q any further - and q ≠ 0.
Then
p = √2 q
and squaring both sides gives
p² = 2q²
This tells us that p² is even, so 2 divides p². But if 2 divides p², it must also divide p, since p is a square number. So p = 2r for some integer r, and we have
(2r)² = 2q²
4r² = 2q²
2r² = q²
We're in the same situation as before - 2 divides q², so 2 divides q. Both p and q are thus divisible by 2, but this contradicts our assumption that p and q are relatively prime, which means √2 cannot be expressed as a rational number p/q. Therefore √2 is irrational.
What you've written in your proof is technically correct, but it doesn't help in the irrationality proof.