How can we say that the method to prove root 2 is irrational is relevent????

2 is not an irrational number.

The method given in NCERT is relevant.

Since, According to the theorem 1.3,

Let p be a prime number. If p divides a², then p divides a, where a is a positive integer.

But here in this case you are using this theorem for 4, which is a composite number. Therefore, the theorem is not applicable.