Let $D(A ,B)= \inf_{\alpha \in A,\beta \in B} d(\alpha ,\beta )$ denote the distance between sets in the power set of a given set where $d(\alpha ,\beta )$ is a metric. Prove that $D(A,B )=0$ doesn't imply $A\bigcap B\neq \varnothing$.
My attempt: Let $D(A,B)= 0$. Then, $\inf_{\alpha \in A,\beta \in B}(d(\alpha ,\beta ))=0$. So there's $a\in A , b\in B$ such that $d(a,b)=0$, which means $a=b$. So, $A\bigcap B\neq \varnothing$. What's the problem in my proof?