# Simple Metric Space Question

Let $(E,d)$ be a metric space and assume that the metric d satisfies $d(x,z) \leq \max(d(x,y),d(y,z))$ for all $x,y,z\in E$.

Prove that if $d(x,y) \neq d(y,z)$ then $d(x,z)=\max(d(x,y),d(y,z))$.

I have spent literally hours on this and I feel like a moron because I can't figure it out. Please help! Thanks!

-
You should enclose all mathematical expressions in \$'s, not just the symbols you can't write in plain text. It makes them much easier to read. Also, I don't think it is good practice to write quantifiers in symbolic form after the quantified statement. – tomasz Sep 24 '12 at 15:27 @Allison Cameron: Have a look at en.wikipedia.org/wiki/Ultrametric_space. Hope it helps. – uforoboa Sep 24 '12 at 15:30 @uforoboa thank you! – Allison Cameron Sep 24 '12 at 15:33 ## 3 Answers Let$d(x,y)\leq d(y,z)\leq d(x,z)$(WLOG) Then,$d(x,z)\leq max(d(x,y),d(y,z))=d(y,z)\implies d(x,z)\leq d(y,z)$Now,$d(y,z)\leq max(d(y,x),d(x,z))=d(x,z)\implies d(y,z)\leq d(x,z)$Both of these$\implies d(x,z)=d(y,z)\implies d(x,z)= max(d(x,y),d(y,z))$- Let$a=d(y,z)$,$b=d(x,z)$,$c=d(x,y)$. Without loss of generality we may assume that$a\gt c$. We have$a\le \max(b,c)$. Since$a\gt c$, we must have$a\le b$. We have$b\le \max(a,c)$, so$b\le a$. It follows that$b=a$. - We already know "$\leq$", so let's assume "$<$" and hope to reach a contradiction. Without loss of generality, let's assume$d(x, y) < d(y, z)$, so that our new assumption is that$d(x, z) < d(y, z)$. But we have$d(y, z) \leq max(d(y, x), d(x, z)) = max(d(x, y), d(x,z))\$, a contradiction to (one of) the above two inequalities.

Further reading: The inequality that your metric satisfies is called the "ultrametric inequality" or the "non-archimedean triangle inequality". A metric space with a metric that satisfies the ultrametric inequality is called an ultrametric space. You can read more about them here: http://en.wikipedia.org/wiki/Ultrametric_space

-