Prove √3 is irrational

Prove by contradiction

Proof:
Suppose √3 is rational
assume √3 = m/n, where m,n are coprime

we square both side of the equation,
3 n^2 = m^2
so m^2 can be divided by 3
implied m can be divided by 3
so we can find integer k, s.t. m = 3k
so 3 n^2 = (3k)^2
n^2 = 3 k^2
so n^2 can divided by 3
implied n can be divided by 3

so we can find m,n both have common factor 3
it contradicts the assumption
so √3 is irrational