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Revision difference : independence of valuations
Version 9 Version 8
Let $|\cdot|_1$, . . . , $|\cdot|_n$ be non-trivial (i.e., they all have also other values than 0 and 1) and pairwise non-equivalent valuations of a field $K$. If $a_1$, ..., $a_n$ are some elements of this field and $\epsilon$ is an arbitrary positive number, then there exists in $K$ an element $y$ which satisfies the conditions Let $|.|_1$, . . . , $|.|_n$ be non-trivial (i.e., they all have also other values than 0 and 1) and pairwise non-equivalent valuations of a field $K$. If $a_1$, ..., $a_n$ are some elements of this field and $\epsilon$ is an arbitrary positive number, then there exists in $K$ an element $y$ which satisfies the conditions
$$|y-a_1|_1 < \epsilon,$$ $$|y-a_1|_1 < \epsilon,$$
$$.$$ $$.$$ $$.$$ $$.$$
$$|y-a_n|_n < \epsilon.$$ $$|y-a_n|_n < \epsilon.$$