If $d(A,B) = \inf_{a \in A, b \in B}{d(a,b)}$ , does $d(A, B) ≤ d(A, C) + d(C, B)? $

Question

I´m trying to prove that if we define the distance between two sets $A, B$ of a metric space $(X,d)$ in the following way:

$$ d(A,B) = \inf_{a \in A, b \in B}{d(a,b)} $$

It verifies that $\phantom{30}d(A, B) ≤ d(A, C) + d(C, B)$

The question seems very easy but I have problems to solve it. Any suggestions?

Answer 1

No, consider $$A=[0,1], B=[2,3], C=[1,2].$$