Prove that any subfield of $\Bbb R$ contains $\Bbb Q$

Question

Prove that any subfield of $\Bbb R$ must contain $\Bbb Q$.

Now for any subfield $F$ of $\Bbb R$, $1\in F$ so, $\Bbb Z \subset F \Rightarrow \Bbb Q \subseteq F$. Have I done it correctly?

Answer 1

You have the right idea. I would add some words explaining why each implication is true, along the lines that $n \in F$ because you can add $n\ 1$'s together, then $-n \in F$ because you can form $0-n$, then $n/m \in F$ because you can divide.

I think most would go the other direction by saying given any $q \in \Bbb Q$, express it as $m/n$, then $m,n \in F$ because...