Let $\displaystyle A=\left\{\dfrac1n:n\in\mathbb N\right\},$ $\displaystyle B=(-1,0]$ and $\displaystyle d(x,y)=|x-y|,$ then find $\displaystyle d(A,B)=\underset{\begin{smallmatrix}

a\in A \\

b\in B

\end{smallmatrix}}{\mathop{\inf }}\,d(a,b)=\underset{\begin{smallmatrix}

a\in A \\

b\in B

\end{smallmatrix}}{\mathop{\inf }}\,\left| \dfrac{1}{n}-b \right|.$

I know it's zero, but don't know exactly how to prove it.

Thanks for the help!