lets see if i could come up with the proof:

let sqrt(2)=m/n, where m,n are positive integers and they are relative prime.

2=m^2/n^2

but as stated before m,n are relatively prime. so n^2=1, therefore m^2=2, which yields m=sqrt(2). that contradicts with m being an integers. therefore sqrt(2) cannot be represented by the quotient of two relatively prime integers. therefore sqrt(2) is irrational.