Question 2. Prove that a nonempty closed subset of R, if it is bounded from below, has a least element.

Let x = (x1, x2, x3) ER³ and suppose you are given a subset UCR³ defined by U = {ER³ : x₂ < 5} Which of the following statements are necessarily true? (i) The complement of U contains a boundary point. (ii) The complement of U contains no interior points. (iii) U is not an open set. (iv) U is not a closed set. (A) (iii) only (B) (i) and (iv) (C) (ii) only (D) (i), (iii), and (iv) (E) (ii) and (iii) (F) (iv) only (G) (i) only (H) (iii) and (iv)

Prove that if A is a nonempty set which is bounded below by 3 then B = {x ∈ R | ∃a ∈ A, b = (1/a)} is bounded above by (1/3).