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I am having trouble seeing the following consequence of the ultrametric inequality, which is supposed to be immediate.

If $|x+y|\leq \max{\{|x|,|y|\}}$, then, equality holds when $|x|\neq |y|$.

I looked up three books/notes and all of them just say that this is immediate.

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The details are given in en.wikipedia.org/wiki/Ultrametric_space#Formal_definition – lhf May 3 '11 at 15:02
    
@lhf: Thanks a lot. That was eating me up. – Milo Kunis May 3 '11 at 15:08

Suppose that $|x|<|y|$ then $|x+y|\le \max\{|x|,|y|\}=|y|$ but in the other hand we have $|y|=|y+x-x|\le\max\{|x+y|,|-x|\}=\max\{|x+y|,|x|\}$ so we have $|y|\le\max\{|x+y|,|x|\}\Rightarrow |y|\le|x+y|$ as $|x|<|y|$ then follows that $|y|=|x+y|$. When you suppose $|x|>|y|$ by exactly the same reason that $|x|=|x+y|$ or just $|x+y|=\max\{|x|,|y|\}$ which finishes the solution.

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