I'm so confused ... you assume that sqrt(N) is a reduced fraction, and so N = p^2/q^2 ... but since (p,q)=1, that implies that N is a rational number, and so we have a contradiction since N is supposed to be an integer greater than 1

tonio

... cuz after that, we are treating N as if it's a rational number, and in that case, we can't really use prime decomposition on it, nor can we use the definition of divisibility ....

Could we do this also:

N is an integer greater than 1 and is not a perfect square.

Assume that sqrt(N) is a rational number say p/q with (p,q)=1 (reduced fraction).

=> N = p^2/q^2 => q^2|p^2 (since N is an integer) => q^2 = 1 (since (p,q)=1) => N = p^2. Contradiction since N is not a perfect square ... so sqrt(N) is not a rational. Since N is positive, sqrt(N) is a real number and since it is not a rational, it must be irrational.