Intersection of $ \Bbb{R}$ and $ \Bbb{Q}_p$
For distinct prime $p$ and $q$,intersection of $ \Bbb{Q}_p$ and $ \Bbb{Q}_q$ is just $ \Bbb{Q}$, but what about $ \Bbb{Q}_p$ and $ \Bbb{R}$ ?
$ \Bbb{R}$ is often said to be completion at infinity, so maybe the intersection should be $ \Bbb{Q}$ as the same as other $ \Bbb{Q}_q$, but how to prove formally this ?
Thank you.