Is the Post Office Metric applicable in $\Bbb{R}^n$ for all $n$? I was required to provide a metric space $(X,d)$ with $x,y\in X$ and $0<r<R$ such that $B_R(x)\subsetneq B_r(y)$. After a lot of thinking and reading, I came by a metric function called the "Post Office Metric", always attributed to $\Bbb{R}^2$, in particular when giving examples for a metric space such as the above. I constructed a metric function (defined on $\Bbb{R}$) of the same concept as follows: $d(x,y)=
\begin{cases}
|x|+|y|,  & \text{if } x\ne y \\
0, & \text{otherwise}
\end{cases}$, 
and to what I checked it seems like a legitimate metric function. Then I looked for it and came by nothing like that, and I begin to feel like I am doing something wrong. I would appreciate your thought on the matter.

The idea is fundamentally similar to Mike F's answer, but perhaps simpler or at least more common in analysis.
Consider $X=[0,1]$ with the induced Euclidian metric. Let $x=0$; then $B_R(x)=[0,x)$. Take say, $R=1/2$.
Then take, say, $y= 1/3$ and $r=2/5$, we have that $B_r(y)=[0,11/15)\simeq [0,0.73)$.
