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Revision difference : independence of valuations
Version 15 Version 14
Let $|\cdot|_1$, \ldots, $|\cdot|_n$ be {\em 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 $|\cdot|_1$, \ldots, $|\cdot|_n$ be {\em 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
\begin{align*} \begin{align*}
\begin{cases} \begin{cases}
|y-a_1|_1 < \epsilon,\\ |y-a_1|_1 < \epsilon,\\
\qquad \vdots \qquad \\ \qquad \vdots \qquad \\
|y-a_n|_n < \epsilon.\\ |y-a_n|_n < \epsilon.\\
\end{cases} \end{cases}
\end{align*} \end{align*}