independence of valuations

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 $K$ , 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

\displaystyle\begin{cases}|y-a_{1}|_{1}<\varepsilon,\\ \qquad\vdots\\ |y-a_{n}|_{n}<\varepsilon.\\ \end{cases}

Title	independence of valuations
Canonical name	IndependenceOfValuations
Date of creation	2013-03-22 14:11:44
Last modified on	2013-03-22 14:11:44
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	22
Author	pahio (2872)
Entry type	Theorem
Classification	msc 11R99
Synonym	approximation theorem
Related topic	TrivialValuation
Related topic	EquivalentValuations
Related topic	WeakApproximationTheorem