For each of the following questions, answer true or false. If true, provide a detailed proof and if false provide a counterexample.

(i) If A Rn is a convex set and B A, then B is a convex set.

(ii) If A Rn is a compact set and B A, then B is a compact set?

(iii) If A Rn is an open set and B Rn is a closed set then AB is closed.

(iv) If A Rn is an open set and B Rn is a closed set then A B is an open set.