Proof of $\mathbb{Q}$ is not cyclic

Solution 1:

This is a proof by contradiction. You want to show $\mathbb Q$ is not generated by $a/b$, i.e., it is absurd that every rational number is an integral multiple of $a/b$. You show its absurdity by observing under this assumption $a/2b$, being a rational number, should be an integral multiple of $a/b$, which it clearly isn't. Hence the assumption that $\mathbb Q$ is generated by $a/b$ cannot be true. Since $a/b$ is arbitrary, this shows $\mathbb Q$ is not generated by any single element, i.e., $\mathbb Q$ is not cyclic.

Solution 2:

The result is referring to the additive group $(\mathbb{Q}, +)$, not the multiplicative group, $(\mathbb{Q}-\{0\}, \times )$. You should be able to work out what is going on from there.