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

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?

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