could you help me? I am trying to prove / disprove that the subtraction of two irrational numbers is irrational
Let $a,b\in\mathbb{R}\setminus\mathbb{Q}$ i want to prove that exists some $i\in\mathbb{R}\setminus\mathbb{Q}$ that satisfies $$a-b=i$$
By contradiction suppose that $a-b=r$ with $r\in\mathbb{Q}$, then
$$a=r+b$$
but $r+b$ is a irrational (i proved it before)
¡¡BUT HERE THERE WAS NO CONTRADICTION!!
In the other hand suppose $ a-b = i $, then $$ a = b + i $$ This is where I have a problem, I know that the sum of a rational number and an irrational number can be rational or irrational. So what can I do?
Thanks in advance:)