independence of valuationsIndependenceOfTheValuations2004-02-25 17:59:092008-12-07 13:13:17Theoremvaluation% this is the default PlanetMath preamble. as your knowledge
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% define commands hereLet $|\cdot|_1$, \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 $K$, all with values real numbers. \,If $a_1$, ..., $a_n$ are some elements of this field and $\varepsilon$ is an arbitrary positive number, then there exists in $K$ an element $y$ which satisfies the conditions
\begin{align*}
\begin{cases}
|y-a_1|_1 < \varepsilon,\\
\qquad \vdots \qquad \\
|y-a_n|_n < \varepsilon.\\
\end{cases}
\end{align*}