1. Field

2. Real Numbers R
The set R of Real Numbers is a field.

3. Example 1
Q is a field.
Z is not a field.
(Why?)

4. Ordered Field

5. Real Numbers R
The set R of Real Numbers is an ordered field.

6. Example 2
C is a field but not an ordered field.

7. Upper (Lower) Bounds
Similar for bounded below, lower bound , and minimum element

8. Example 3 Determine which of the following sets are bounded above.
Determine which of the following sets have a maximum element.

9. Example 3 (a)
Analysis
Solution

10. Example 3 (b)
Analysis
Solution

11. Example 3 (c)
Analysis
Solution

12. Archimedean Property

13. Density Property

14. Least Upper Bounds
Similar for greatest lower bound, and infimum

15. Equivalent Statement
Similar for greatest lower bound, and infimum

16. Example 4
Show that
Analysis
Solution

17. Example 5
Determine the supremum of
Analysis
Solution

18. The Completeness Axiom