The main question: Let S be a subset in Rn which is both open and closed. If S is non-empty, prove that S= Rn. I am allowed to assume Rn is convex. Things I've considered and worked with: The compliment of Rn is an empty set which has no boundaries and therefore neither does Rn. Therefore there exists NO points P such that the delta neighborhood of P is located within Rn and its compliment. This means it is open, because any d-nbhd within Rn is a subset of Rn, and it is closed because all of the boundaries of Rn are within the set. Therefore Rn is both open and closed. Thats where I am stuck. And also, I am not sure If all of that is valid logic. The second question I had, as I have two questions remaining on my assignment after working on it for 10 hours today and yesterday another 10, and becoming quite desperate for help is: "Let f(x,y) be defined on the square -1 <= x <= 1, -1<=y<=1 as follows: f(x,y) = 1 for all (x,y) where Abs(x^3) <y < x^2 and f(x,y) = 0 otherwise. Show that f(x,y) approaches 0 as (x,y) approach zero on any straight line through the origin. Determine if the lim (x,y) -> 0 exists."