Infering a bound away from another point

Question

Suppose that $x,y,z \in (X,d_X)$, some metric space $(X,d_X)$, each point being distinct. Suppose that I also know that $d(x,y)>\epsilon_1$ and that $d(y,z)>\epsilon_2$ and $0 < \epsilon_1<2^{-1}\epsilon_2$...then I infer $$ d(x,z) \geq \left| d(x,y) - d(y,z) \right|, $$ but how can I conclude that this bound is strictly positive?

Answer 1

You can't conclude that. When the only condition is that the given distances are greater than $\varepsilon_1$ and $\varepsilon_2$, resp., then all conditions on $\varepsilon_1, \varepsilon_2$ are irrelevant, you could always have the case

$$d(x,y)=d(y,z)=\max(\varepsilon_1, \varepsilon_2)+1.$$

The same is true if you replace in the conditions on the distances ">" with "<", then you can't exclude the possibility that both distances are equal to $\frac12\min(\varepsilon_1, \varepsilon_2)$.

In order to make sure that $|d(x,y)-d(y,z)|$ is positive, you need to be able to bound one distance from above and the other from below, like for example

$$d(x,y) > \varepsilon;\; d(y,z) < \frac12\varepsilon,$$

from which $|d(x,y)-d(y,z)| > \frac12\varepsilon$ follows.