MA180/186/190 Calculus Semester 2. Week 9: The Maximum, Supremum, Minimum, Infimum and the Completeness Axiom
Welcome to Week 9.
We will finish working on Chapter 2 this week. In Lecture 16, we will discuss the Axiom of Completeness for bounded subsets of ℝ, and look at an essential difference between the set of rational numbers and the set of real numbers. If a set of real or rational numbers is bounded (above), its elements to not persist indefinitely as we move along the number line in the positive direction, we eventually pass all of them. We can consider whether the set has a greatest element or maximum. It may not, for example the open interval (0,1) does not have a maximum element. While a bounded set of real numbers may not have a maximum element, it must have a supremum or least upper bound, which is a real number. It is not true that every bounded set of rational numbers has a supremum that is a rational number. This distinction between the real and rational numbers is related to the Axiom of Completeness for the real numbers, which is our theme for Wednesday's lecture.
Here is Lecture 16 (Wednesday April 14).
On Thurdsay we will have some general advice about the calculus exam questoins, and a look at some exam-type questions relating to Chapter 2.
Here is Lecture 17 (Thursday April 15).
Slides/notes from Lecture 17. Some more advice on this exam question can be found at the end of the Chapter 2 lecture notes.
Relevant sections of the lecture notes this week are Sections 2.3 in Chapter 2 and Section 3.1 in Chapter 3. There are more examples in the lecture notes than we will discuss in our lectures, and more detailed explanations in some places.
Weekly Problem 9
The weekly problems are just for fun. They have nothing much to do with our curriculum. Please send me an email if you have a solution that you would like to share with the class!