(Ir)rationality of Real Numbers I am working on this proof and this is what I got so far, can someone help me verify if what I have done is right?

For all real numbers $x$ and $y$, if $x+y$ is rational and $x-y$ is irrational
  then $x$ is irrational and $y$ is irrational.

Proving the contrapositive and since contrapositive is logically equivalent to the original statement, we can conclude the original statement.
Contrapositive: For all real numbers $x$ and $y$, if $x$ is rational or $y$ is rational, then $x+y$ is irrational or $x-y$ is rational.
Proof: Suppose $x$ and $y$ are real numbers such that $x$ is rational. Then let $y = x$. We want to prove that $x+y$ is irrational or $x-y$ is rational.
Since $x$ is rational, we know that $x = a/b$ where $a$ and $b$ are integers and $b \not= 0$.
and $y = -x$
then $x-y = x-x = 0$, since $0$ is integer, and it is rational, that implies that $x-y$ is rational. End of Proof.
Now, can someone help me prove the original statement? Without trying with contrapositive?

You can proceed directly as follows: $2x = (x+y) + (x-y)$ which must be irrational as it is the sum of a rational and an irrational. So $x$ is irrational. Similarly $2y = (x+y) - (x-y)$ is irrational.
