Justification for the inclusion of triangle inequality in the definition of a metric How does one justify that triangle inequality must be included in the definition of a metric?
Well I know the geometric reason that it was motivated from seeing the geometry in R2,R3 etc,but are there any other reasons?
For example if I'm teaching a student who is seeing metrics for the first time how do I justify it to him that triangle inequality must be included in the definition other than geometry?
Any insights or thoughts regarding this would be helpful.
Is it something got to do with the length of shortest path or something?

Wikipedia gives an intuitive meaning of each of the three axioms that define a metric space:

Ignoring mathematical details, for any system of roads and terrains
the distance between two locations can be defined as the length of the
shortest route connecting those locations. To be a metric there
shouldn't be any one-way roads. The triangle inequality expresses the
fact that detours aren't shortcuts. If the distance between two points
is zero, the two points are indistinguishable from one-another.

