= Weak approximation for inequivalent absolute values
{title2=$K\text{ is dense in }\prod_{i=1}^r K_{v_i}$}
= Weak approximation theorem
{synonym}
For finitely many pairwise inequivalent nontrivial <absolute values on a field>, an element of the field can approximate independently prescribed targets in the associated <completion of a valued field> at each place to arbitrary positive accuracies. Equivalently the diagonal map has dense image in the finite product. Requiring finitely many inequivalent places is essential; the conclusion controls finitely many inequalities, not infinitely many independently prescribed bounds at once, or incompatible targets at one place. <Weak approximation for number fields> is the class-field-theoretic instance.
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