Subgroup of $(\mathbb Q, +)$ with finite index

Question

I don't understand a step in the proof here

Why must it be that $nq\in H$ ?

Answer 1

If $G$ is a group of order $n$, then $g^n=1$, for every $g\in G$ (multiplicative notation).

If we apply this to the quotient group $G/H$, where $H$ is a normal subgroup of $G$ and $[G:H]=n$, then, for $g\in G$, $$ H=(gH)^n=g^nH $$ and so $g^n\in H$.

In additive notation, $ng\in H$.