Let (F, <) be an ordered field, let S be a nonempty subset ofF, let c ? F, and for purposes of this problem let cS = {cx | x ?S}. (Do not use this notation outside this problem without definingwhat you mean by the notation.) Assume that c > 0.
(i) Show that an element b ? F is an upper bound for S if andonly if cb is an upper bound for cS.
(ii) Show that S has a least upper bound if and only if cS has aleast upper bound, and that (when these sets have least upperbounds), they are related by
l.u.b.(cS) = c · l.u.b.(S).
