The problem: Come up with an example of sets 
$A$, and $B$ in $\mathbb{R}^2$ such that $A \subset B$, $A \neq B$, and the boundary points of $A$, $bd(A) = B$. 

Here is what I think. I was going to set $A = \{ \bar{u} \in \mathbb{R}^2 | \|u \| =1 \} $ and $B = \{ \bar{u} \in \mathbb{R}^2 | \| u \| \leq 1 \}$ but I cannot think of good examples. 

Can someone give me some hints please? Try not to solve the problem!

Thank you very much!!