Positive Distance between two sets in a metric space

Question

Let $(X,d)$ be a metric space, $A$ and $B$ be subsets of $X$ so that $d(A,B)>0$ where $d(A,B)=\text{inf}\;\{d(a,b): a \in A,b \in B\}$.

How to prove there exist open sets $U$ and $V$ so that $A \subset U$, $B \subset V$ and $U \cap V=\phi$ ?

I don't know where to start. Any hint?

Answer 1

Hint: At each point in $A$ and then at each point in $B$, consider a ball of radius $d(A,B)/2$. What do you notice about this construction?