Proof regarding irrational and rational numbers

Prove that the difference of any irrational number and any rational number is irrational.

Can someone please help me with this proof? Thanks.

Let x be an irrational number and let y be a rational number.
Then we can express y in terms of the integers p and q such that
y = p/q
Further, because x is irrational, there exist NO integers r and s such that x = r/s

Now suppose that there exist two integers m and n such that
x - y = m/n
That is
x - p/q = m/n =>
x = (m/n) + (p/q) = (mq + np)/nq
= r/s
where, since the set of integers is closed under addition, subtraction and multiplication,
r = (mq + np) and s = nq are integers
This implies that x is a rational number. A contradiction.
Hence, the difference between an irrational number and a rational number must be an irrational number.

Corollary: The addition of an irrational number and a rational number is irrational.

Easy.

Suppose you have a number x such that x a rational is a rational. Then
x - a/b = c/d, where a, b, c and d are integers.

Then x = c/d a/b = (cb ad)/(bd), showing that x must be rational. So the difference of an irrational and a rational can't be rational and must be irrational.