Q: If $p,q$ are different primes, show the embedding $$\mathbb{Q} \rightarrow \mathbb{Q}_q \times \mathbb{Q}_p$$ $$x \rightarrow (x,x)$$ is dense in the product space $\mathbb{Q}_q \times \mathbb{Q}_p$.
Furthermore, show the embedding $$\mathbb{Q} \rightarrow \mathbb{R} \times \mathbb{Q}_p$$ $$x \rightarrow (x,x)$$ is dense in the product space $\mathbb{R} \times \mathbb{Q}_p$.
My idea: The professor gives us a hint to use Chinese Remainder Theorem. However, I still have no idea to start off my solutions...Can anyone help to explain how to do this proof? I am confused about this topic also. Thank you.
Let's look at the second one. Given $a\in \Bbb R$, $b\in\Bbb Q_p$ and $\newcommand{\ep}{\varepsilon}\ep>0$ one needs to prove the existence of $x\in\Bbb Q$ with $|a-x|_\infty<\ep$ and $|b-x|_p<\ep$. There are certainly $a'$, $b'\in\Bbb Q$ with $|a-a'|_\infty<\ep/2$ and $|b-b'|_p<\ep/2$ so it suffices to prove that there is $x\in\Bbb Q$ with $|x-a'|_\infty<\ep/2$ and $|x-b'|_p<\ep/2$. Let $y=x-b'$ and $c=a'-b'$. Then we want $y\in\Bbb Q$ with $|y-c|_\infty<\ep$ and $|y|_p<\ep/2$.
To achieve this, consider $u=p^N/(1+p^{2N})$ where $N$ is a sufficiently large integer. Choosing $N$ large enough, gives $|u|_p<\ep/4$ and $|u|_\infty<\ep/4$ say. Then $|mu|_p<\ep/4$ for all integers $m$. We can now take $m\in\Bbb Z$ so that $mu$ lies in the interval $(c-\ep/4,c+\ep/4)$ inside $\Bbb R$.