Real Analysis Sets example 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!!

Your example does not work, since $(0,0)$ is not a boundary point of B, just take the ball of radius $\frac{1}{2}$ around it, which clearly does not contain any elements from $A$. You need something dense in $B$ to be your set, and there is nothing quite like $\mathbb{Q}$ as far as dense sets go. Can you finish it from here? 
