# Is it always possible to define an absolute value in an ordered field?

I am trying to show (not sure if possible) that i can generalize all basic arithmetic operations between limits of sequences of real numbers to any ordered field, so i need to build a generalized notion of absolute value. The reason why i am asking for an absolute value instead of a norm is because when i try to prove that $\lim c.a_n=c.\lim a_n$ (where c is a constant of F) i need to be able to use this formulae: $$n(k.a_n - k.L)=|k|.n(a_n-a)$$ (where n() is the norm function and || is the absolute value i'm looking for). So the question is can i use the same criteria for defining an absolute value in any ordered field that is used for defining absolute value in real numbers? If the answer is yes, will this absolute value do the job or is there some any subtle detail i have to take care of? Thanks in advance

• I don't know what an absolute value is. If it is defined in the usual way then it exists in general. But limits get tricky. Do you want the limit of the sequence $(1/n)$ to be $0$? Then in a non-standard model of analysis, the sequence $(c/n)$ may not have a limit. – André Nicolas Feb 11 '16 at 22:18
• Maybe you might want to look at the notion of Metric space en.wikipedia.org/wiki/Metric_space. It generalizes idea of distance and also the limit. – Sil Feb 11 '16 at 22:25