approximation theorem

《Algebraic Number Theory》 · 第 117 页 · Chapter II，Section 3「Valuations」，Theorem 3.4

(3.4) Approximation Theorem. Let $|\ | _1,\ldots,|\ | _n$ be pairwise inequivalent valuations of the field $K$ and let $a_1,\ldots,a_n\in K$ be given elements. Then for every $\varepsilon>0$ there exists an $x\in K$ such that $|x-a_i|_i<\varepsilon$ for all $i=1,\ldots,n$.