I am looking for an enlightening proof of the following fact:
Let $F(t)\in \mathbb{Z}[[t]]$ and suppose $S$ is a finite set of places on $\mathbb Q$ containing $\infty$. If for every $v\in S$ $F(t)$ is meromorphic of radius $r_v$ in $\mathbb Q_v$ and $\prod_{v\in S} r_v>1$ then $F(t)\in \mathbb Q(t)$ (is rational over $\mathbb Q$).