18.099b Problem Set 2b Due: Thursday, February 26th (in class or before).

1. Suppose F ⊆ ℝ is a closed subset of the real numbers for which sup F exists. Show that sup F ∈ F.

2. Suppose E ⊂ X is a subset of a metric space. Let E′ be the set of all limit points of E. Show that E′ is closed. (Hint. Show that if x is a limit point of E′ then x is a limit point of E)