I want to show that with p-adic ultrametric : $|.|_{p}=p^{-ord(.)}$ where . is a integer and p is a prime , with $|0|_{p}=0$ if we have balloons $$B(a,R)=\{z\in \mathbb{Z}||z-a|_{p}\le R\}$$ $$B(b,R)=\{z\in \mathbb{Z}||z-b|_{p}\le R\}$$
then they can never overlap.
Proposition: They can never overlap without being the same.
Assume we have two different points y and y' in one ball, if $y\in B(a,R)$ with the strong triangle inequality $|a+b|_{p}\le sup\{|a|_{p} ,|b|_{p}\}$
now set: $a:=z-y'$ and $b:= y'-y$ so we get : $|z-y|_{p}\le sup\{|z-y'|_{p}. |y'-y|_{p} \}$
So that means $|z-y|_{p} \le R \Leftrightarrow |z-y'|_{p}\le R $ e.g. every point of the ball is a center point so the balloons can never overlap without being the same.
Is this a proof for the proposition? (I never made use of the second balloon...)