Open balls in normed space Let $X$ be a normed space. Let the open ball $B(x,r_1)$ in $X$ be contained in the open ball $B(y,r_2)$. If $r_1=r_2$, how can I show that $x=y$? Geometrically, it is ok for me but how to write a mathematical proof of it?

Suppose $x \neq y$. Note that $x \in B(y,r)$ so $\|x-y\|<r$. If $0<t<\frac r {\|x-y\|}$ and $t >\frac r {\|x-y\|}-1$ then $x-t(y-x)$ belongs to the first ball but not the second. [Check that such a number $t$ exists!].
