$0\neq a\in \mathbb Q, b\in \mathbb R \setminus\mathbb Q \text{ (b is irrational)}$ Prove that $\frac a b$ is irrational.
From defintion $a=\frac m n$ such that $m,n\in \mathbb Z, n\neq 0$.
Take the contrapositive: suppose $\frac m {nb}\in \mathbb Q$ prove $\frac m n\notin \mathbb Q$.
Immediate contradiction from defining $m,n\in \mathbb Z, n\neq 0$. Thus $\frac m {nb}$ is irrational.
Well I'm not sure I'm using the contrapositive right and I tried to combine contrapositive with proof by contradiction but I have a feeling I'm wrong...