4. Use the least upper bound property of R to prove that every nonempty subset of R that is bounded below has an infimum.

Let A CR be a nonempty subset which is bounded below, and x E R<0. Show that the scaled set xA is bounded above, that its supremum in R exists and satisfies sup(xA) = xinf(A).

a)To prove this directly, you'd suppose that A is a subset of B.  Then you'd need to show that sup(A)≤sup(B)sup(A)≤sup(B).  Since A is bounded above, sup(A) is the least upper bound for A which, by definition of l.u.b., is less than or equal to all upper bounds for A.   b)Note that, max{sup(A), sup(B)} is either sup(A) or sup(B).  So you have a couple of natural cases to break things down into.  c) I am not sure of

Let A be a nonempty subset of R that is both bounded above andbelow and let B be a nonempty subset of A. Prove or disprove eachof the following assertions.(a) inf(A) ≤ inf(B) ≤ sup(B) ≤ sup(A).(b) If inf(A) = sup(A), then A has exactly one element.(c) If inf(A) = inf(B) and sup(A) = sup(B), then A = B.