Proposition 1   Let $p_1,\ldots,p_n \in \Ints$ be distinct prime numbers and let $\mid\cdot\mid_{p_i}$ be the corresponding $p$ -adic valuations of $\Rats$ . Let $a_1,\ldots,a_n\in \Ints$ and let $\epsilon_i$ be arbitrary positive real numbers, then there exists $y\in\Ints$ such that for all $i=1,\ldots,n$ : $$\mid y - a_i \mid_{p_i} < \epsilon_i$$

Proof. Let $p$ be an arbitrary prime, and let $\epsilon$ be an arbitrary positive real number. Notice that $\Ints$ injects into $\Ints_p=\varprojlim \Ints/p^n\Ints$ , the $p$ -adic integers. For any $b\in \Ints$ , we also write $b$ for its image in $\Ints_p$ , and it can be written as a sequence $b=(b_j)$ with $b\equiv b_j \mod p^j$ . Let $n=n_{p,\epsilon}\in \Nats$ be such that $p^{-n} < \epsilon$ (and thus for any other $c\in \Ints $ such that $c\equiv b_n \mod p^n$ we have $\mid b-c \mid_p \leq p^{-n} < \epsilon$ ).

Now, for the proof of the proposition, let $n_i=n_{p_i,\epsilon_i}$ and recall that by the Chinese Remainder Theorem we have an isomorphism:

$$\prod_{i=1}^n \Ints/p_i^{n_i}\Ints \equiv \Ints/(\prod p_i^{n_i})\Ints$$

Therefore we can find an element $\tilde{y}$ of $\Ints/(\prod p_i^{n_i})\Ints$ (and thus a lift $y$ of $\tilde{y}$ to $\Ints$ ) such that $y \equiv a_i \mod p_i^{n_i}$ for all $i=1,\ldots,n$ . Hence: $$\mid y - a_i \mid_{p_i} < \epsilon_i$$