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If $p,q$ are distinct primes, it is true that the subset $\mathbb{Z} \times \mathbb{Z}$ is dense in $\mathbb{Z}_{p} \times \mathbb{Z}_{q}$. However, is it true that $\lbrace (x,x), x\in \mathbb{Z} \rbrace $ is dense $\mathbb{Z}_{p} \times \mathbb{Z}_{q}$? I think I need to prove the fact I must prove or disprove that my set is dense in $\mathbb{Z} \times \mathbb{Z}$? (The topology here is the product topology, where each toology is induced by the usual p adic metric)

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Individually, $\mathbb{Z}$ is dense in $\mathbb{Z}_p$ and $\mathbb{Z}_q$ since that's how we get $p$- and $q$-adic numbers by completion.

Even more we can solve $x \equiv a \mod p^k$ and $x \equiv \mod q^k$ simultaneously for any $p,q$ and any $k$ by the chinese remainder theorem or pigeonhole principle.

It works for any number of primes, simultaneously. This is called weak approximation:

Theorem 4.2

Let $K$ be a field and $|\cdot|_1, \dots,|\cdot|_n$ be pairwise inequivalent $a_1, \dots, a_n \in K$ and $\epsilon_1, \dots, \epsilon_n \in \mathbb{R}^+$. There exists an $x \in K$ satisfying $|x - a_i|_i < \epsilon_i$ simultaneously.

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