In general, you can prove this:
Let be a metric space, and be a increasing positive concave function with , roughly,
- on
- on .
Then is a metric space, where defined by for .
The first condition is required for M1, and the last two is for M3,
you can see that M2 follows directly without any condition.
This was what I have proved when I was 2nd year BC.
You can also show what if .