Let be a compact metric space, which metric is .
Let be p and q two points of .
Can I say that, since is compact, the distance between p and q is finite and reachable?
The distance map is a continuous map from a compact space (compact because X is, and continuity requires some proof if you haven't already proved this).
But the continuous image of a compact set is compact which means closed and bounded in so the distance between the two points is definitely finite. I am not sure what you mean by reachable.
Yeah you are just talking about the extreme value theorem then I think.
Basically here is the deal. The image of the space is compact in which is complete. So we know the image is closed an bounded. In particular the inf and sup of this set would be a limit point, but it is closed, so it contains all of its limit points. Thus is achieves both its max and min? Sound good?