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.
$\endgroup$ 21 Answer
$\begingroup$$\endgroup$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.
