NOTE: we know that √2 is irrational.

This is how I did it. I just want to make sure that if I did it right or not.

Proof by Contrapositive:

Suppose √2 + √n = r, a rational number

Then √n = r - √2

We square both sides

n = (r - √2)²

expand the right hand side

n = r² - 2*r*√2 + 2

therefore, √2 = (n – 2 - r²) / (-2*r)

numerator and denominator are rational, so √2 is rational, which is not true. Therefore √2 + √n is irrational.

if I am wrong, pls provide me with right proof.

one more question...

give an example of a positive irrational number, x, so that √2 + x is rational.