Problem 4. Show that the first definition of continuity implies the second one. 1. Definition 1. A preference relation >= (this is curly, nor greater than or equal to) on X =Rn is continuous if for any two sequences {x_k}k=1,2,... and {y_k}k=1,2,... of X such that lim k {x_k} = x* and limk {y_k} = y* , if x_k >= y_k for each k, then x* >= y* .

2. Definition 2. A preference relation >= on X = Rn is continuous if for any x X, U(x) = {y X : y >= x} and L(x) = {y X : x >= y} are closed sets. Problem 5. Suppose >= on X = R2 + is rational, continuous, and monotonic.

1. Show that for any x X, I(x) = U(x) L(x) = {y X : x y} is closed.

2. Show that for any x X, there exists R such that x (, 2 ).