Prove that any two nontrivial subgroups of $\mathbb{Q}$ have nontrivial intersection

I need to prove that any two nontrivial subgroups of $\mathbb{Q}$ have a nontrivial intersection as part of a larger proof that $\mathbb{Q}$ cannot be represented as a nontrivial direct product.

(Yes, I realize that there are other ways to prove that $\mathbb{Q}$ cannot be represented as a nontrivial direct product, but I am not interested in those, and will not accept answers showing me how to do that).

Based on the density of the rationals, I guess this idea kind of makes sense, but I am doing this proof for an abstract algebra class, and I don't think real analysis proofs would be accepted.

Everywhere I've looked online has said this is "trivial" or "easy to see" without bothering to actually prove it for those of us mere mortals who don't see it as such. As for myself, I have literally no idea where to even get started proving this. Could somebody help me figure this out (without giving me cutesy examples that aren't really all that related with the expectation that I should be able to figure it out from looking at those - that's not really how my brain works)?

Thank you.

Say you have two non-trivial subgroups $G,H$. Take non-zero $\frac ab\in G, \frac cd\in H$ where $a,b,c,d$ are integers. Then, by closure of addition, we have $a\in G,c\in H$, and by applying closure of addition again, we get $ac\in G\cap H$.