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Preview
Preview
We’ve now defined the set of real numbers by adding the Completeness Axiom to the field and order axioms. That axiom is all about suprema, so today we’ll discuss that concept further and prove a few results we’ll use in later investigations in the course. (Doing so will also give us practice with the four proof methods discussed in earlier lessons.) We’ll also further explore R, its properties, and the decimal representation of real numbers.
In Module I we’ll establish several results related to suprema. In Module II we’ll prove that we can always bound a real number above by an integer. Then, in Module III we’ll refine this and discuss how we can always bound a real number above and below by rational numbers. This will lead to a Fundamental Theorem of Arithmetic-like result: each real number can be expressed (or approximated) by a decimal.
Review
Review
Learn
Learn
Work through the lesson notes below. In class, I will fill in the "See Class Notes" boxes. Optional: Watch the video(s) below the embedded notes to get another instructor's take on the content covered in the notes.
Some tips for you as you work through these resources:
I recommend using Cornell Notes (or a modification of it; see this video starting at the 1:05 mark) to take notes on the lesson and the videos. This note-taking method balances detail with big-picture thinking to help you summarize and retain what you are learning. See this other video for additional note-taking techniques you might want to experiment with.
Lesson Notes
Lesson Notes
Lesson 4.pdfReflect
Reflect
If you are currently enrolled in this course with me, submit the written reflections Google Form I have emailed you after working through the lesson notes and videos. Some tips:
Submit substantive, but concise, answers to each question; you will be doing the future you a big favor by taking time now to accurately and succinctly summarize what you have learned from the lesson.
Send yourself a copy of your reflections; they will come in handy later when you start preparing for quizzes and other assessments.
If you are not currently enrolled in this course with me, those written reflections ask three reflective questions designed to help you retain what you've learned and pinpoint any remaining areas of confusion. Those questions are:
Please summarize the main mathematical takeaways from the lesson notes.
What was the most interesting part of what you learned, and why?
What, if anything, do you still find confusing?
Practice
Practice
Work through the practice problems suggested below to see how much of this lesson you've understood. Book problems come from the book for the course, Mathematical Analysis, Second Edition, by Tom Apostol.
Lesson 4 PP.pdfPage updated
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