Could you please confirm if this proof is correct?
Theorem: If $q \neq 0$ is rational and $y$ is irrational, then $qy$ is irrational.
Proof: Proof by contradiction, we assume that $qy$ is rational. Therefore $qy=\frac{a}{b}$ for integers $a$, $b \neq 0$. Since $q$ is rational, we have $\frac{x}{z}y=\frac{a}{b}$ for integers $x \neq 0$, $z \neq 0$. Therefore, $xy = a$, and $y=\frac{a}{x}$. Since both $a$ and $x$ are integers, $y$ is rational, leading to a contradiction.
It's wrong. You wrote $\frac{x}{z}y = \frac{a}{b}$. That is correct. Then you said "Therefore $xy = a$. That is wrong.
You need to solve $\frac{x}{z}y = \frac{a}{b}$ for $y$. You get $y = \frac{a}{b} \cdot \frac{z}{x}$.
As I mention here frequently, this ubiquitous property is simply an instance of complementary view of the subgroup property, i.e.
THEOREM $\ $ A nonempty subset $\rm\:S\:$ of abelian group $\rm\:G\:$ comprises a subgroup $\rm\iff\ S\ + \ \bar S\ =\ \bar S\ $ where $\rm\: \bar S\:$ is the complement of $\rm\:S\:$ in $\rm\:G$
Instances of this are ubiquitous in concrete number systems, e.g.
You can directly divide by $q$ assuming the fact that $q \neq 0$.
Suppose $qy$ is rational then, you have $qy = \frac{m}{n}$ for some $n \neq 0$. This says that $y = \frac{m}{nq}$ which says that $\text{y is rational}$ contradiction.
A group theoretic proof: You know that if $G$ is a group and $H\neq G$ is one of its subgroups then $h \in H$ and $y \in G\setminus H$ implies that $hy \in G\setminus H$. Proof: suppose $hy \in H$. You know that $h^{-1} \in H$, and therefore $y=h^{-1}(hy) \in H$. Contradiction.
In our case, we have the group $(\Bbb{R}^*,\cdot)$ and its proper subgroup $(\Bbb{Q}^*,\cdot)$. By the arguments above $q \in \Bbb{Q}^*$ and $y \in \Bbb{R}\setminus \Bbb{Q}$ implies $qy \in \Bbb{R}\setminus \Bbb{Q}$.
Let's see how we can modify your argument to make it perfect.
First of all, a minor picky point. You wrote $$qy=\frac{a}{b} \qquad\text{where $a$ and $b$ are integers, with $b \ne 0$}$$
So far, fine. Then come your $x$ and $z$. For completeness, you should have said "Let $x$, $z$ be integers such that $q=\frac{x}{z}$. Note that neither $x$ nor $z$ is $0$." Basically, you did not say what connection $x/z$ had with $q$, though admittedly any reasonable person would know what you meant. By the way, I probably would have chosen the letters $c$ and $d$ instead of $x$ and $z$.
Now for the non-picky point. You reached $$\frac{x}{z}y=\frac{a}{b}$$ From that you should have concluded directly that $$y=\frac{za}{xb}$$ which ends things, since $za$ and $xb$ are integers.
I don't think it correct. It seems like a good idea to indicate both x as an integer, and z as a non-zero integer. Then you also want to "solve for" y, which as Eric points out, you didn't quite do.
$$a\in\mathbb{Q},b\in\mathbb{R}\setminus\mathbb{Q},ab\in\mathbb{Q}\implies b\in\mathbb{Q}\implies\text{Contradiction}\therefore ab\not\in\mathbb{Q}.$$
a is irrational, whereas b is rational.(both > 0)
Q: does the multiplication of a and b result in a rational or irrational number?:
Proof:
because b is rational: b = u/j where u and j are integers
Assume ab is rational:
ab = k/n, where k and n are integers.
a = k/bn
a = k/(n(u/j))
a = jk/un
before we declared a as irrational, but now it is rational; a contradiction. Therefore ab must be irrational.
