Prove there are no injective homomorphisms

Prove that there are no injective homomorphisms from the additive groups $\mathbb{Q}\to \mathbb{Z}$.

If there were such an injective homomorphism, say $\psi$, then $Ker(\psi )=\{e\}$.

Notice, $\psi (1)=1$.

Let $\varphi (\frac{1}{2})=a$. Then $2a=\varphi (\frac{1}{2})+\varphi (\frac{1}{2})\varphi (\frac{1}{2}+\frac{1}{2})=\varphi (1)=1$. Then we have it so $a=\frac{1}{2}$. However, no such element exists in $\mathbb{Z}$.