Let be a metric space and a function with , , for . Prove that is a metric on .

*Solution*. The function is increasing and concave down. Let

- Since then (because is increasing), so . Of course, for (because is increasing) and because .
- Symmetry is easy because is so.
- (because is increasing) (because and is concave down), so triangle inequality is verified.

**Remark**. You want to prove (then you can substitute , ).

By concavity,

and

.

Sum them up, you are done.