Additive group of rationals has no minimal generating set

Let $S$ be a minimal generating set of $\mathbb{Q}$, and take $a\in S$. Let $H\leq \mathbb{Q}$ be the subgroup generated by the elements of $S$ which aren't $a$. Because $S$ is minimal, $H$ is a proper subgroup of $\mathbb{Q}$. Define $G:=\mathbb{Q}/H$. Then $G$ is non-trivial and cyclic (it is generated by the class of $a$), so $G\cong\mathbb{Z}/n\mathbb{Z}$ or $G\cong \mathbb{Z}$. This is a contradiction, because every element of $\mathbb{Q}$ is divisible by $n$, so the same should be true of a quotient of $\mathbb{Q}$.

Let $S\subseteq\mathbb Q$ be such that $\langle S\rangle=\mathbb Q$. Fix $a\in S$, and put $T=S\setminus\{a\}$, let us see that also $\langle T\rangle=\mathbb Q$. We have $$\frac{a}{2}=a\cdot k_0+\sum_{i=1}^na_i\cdot k_i,$$ for some $k_i\in\mathbb Z$ and $a_i\in T$. Then $$a=a\cdot (2k_0)+\sum_{i=1}^na_i\cdot (2k_i),$$ that is, $$a\cdot m=\sum_{i=1}^na_i\cdot (2k_i),$$ where $m=1-2k_0$ is nonzero; as $k_0$ is an integer.

Now $\frac{a}{m}$ can be expressed as a combination of elements of $S$, say $\frac{a}{m}=a\cdot r_0+\sum_{i=1}^lb_i\cdot r_i,$ with $b_i\in T$,$r_i\in\mathbb Z$, thus $$a=a\cdot mr_0+\sum_{i=1}^lb_i\cdot mr_i=\sum_{i=1}^na_i\cdot r_0(2k_i) +\sum_{i=1}^lb_i\cdot mr_i.$$