I'm taking Real Analysis as well, so my answers may not be correct.

For the intersection, you can say take a x, and then you have the set E_x. Take any element e \in E_x as your new x, and you get a new set. You can continue this and by Archimedean property the smallest such element would be {0}, so their intersection would be {0}. You can probably do the same with the other side.