Equality between distribution functions on rationals
By definition of a distribution function $F$ and $G$ are right-continuous. Let $x\in \Bbb R$. There exists a sequence $(q_n)_{n\in \Bbb N}$ such that $q_n \in \Bbb Q$, $q_n \geq q_{n+1} \geq x$ and $q_n \to x$. Since $F$ and $G$ are right-continuous it holds $$F(x) = \lim_{n\to\infty}F(q_n) = \lim_{n\to\infty} G(q_n ) = G(x)$$ for every $x\in \Bbb R$. We have used that $F(q_n) = G(q_n)$ for every $q_n$.