Mathematics 242 – Principles of Analysis Discussion 2 – Completeness of R, continued September 8, 2000

Background

In class on Wednesday, we introduced the L.U.B., or Completeness Axiom for the real number system:

Completeness (LUB) Axiom. Every nonempty set of real numbers that is bounded above has a least upper bound (lub) in the real numbers.

Today, we want to begin to explore some of the consequences of this innocuous-seeming but extremely important property.

Discussion Questions

A) Identify a lub for each of the following nonempty sets of real numbers. You may use facts you know from calculus if needed.

1) A = {x | x^{2} − 3x − 6 ≤ 0}

2) B = \{\frac{2n}{n+3} | n ∈ N\}

3) C = { cos x | 0 < x ≤ π/2}

B) In this problem, you will show that if A is a set of real numbers that is bounded above and nonempty, then lub(A) is unique – we are actually justified in saying “the least upper bound” instead of “a least upper bound”. To show that something is unique, a standard approach is to assume you have any two of them, then show that the two must be equal. So suppose that α and α′ are both least upper bounds of A. Then … (you complete the proof). (Hint: Go back to our original definition of a lub and use what that says.)

C) For sets that are bounded below there is a corresponding notion of greatest lower bound.

1) Give a precise definition of what it should mean for β ∈ R to be a greatest lower bound for a set B that is bounded below.

2) Prove that the Completeness Axiom above implies that every nonempty subset of the real numbers that is bounded below has a greatest lower bound. (Hint: The best way to think about this is that you can deduce what you want from the LUB axiom if you can “turn lower bounds into upper bounds” somehow – for instance by doing some operation to your set that reverses all inequalities.)

D) For each of the following statements that is true, give a short proof; for those that are false, give a counterexample.

A) Suppose α = lub(A). Then |α| = lub({|x||x ∈ A}). B) Suppose α = lub(A) and β = lub(B). Then α + β = lub({x + y|x ∈ A, y ∈ B}. Assignment

Group writeups due in class, Wednesday, September 13.