We define the diameter of a nonempty subset A of M by { }. Show that A is bounded if and only if diam(A) is finite.
this is seems obvious, if diam(A) is not finite, then the distance from a to b can be infinite, which says that A is not bounded.
However I cannot think of an official proof of this, can anyone help?
A is bounded if there is some and some constant C<infinity s.t for all
So if We know A is bounded, we have and which implies
so this shows that diam(A) is less than infinity, hence its finite. Is this correct?
How about prove from the other direction?