Field

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

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

Ordered Field

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

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

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

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

Example 3 (a) Analysis Solution

Example 3 (b) Analysis Solution

Example 3 (c) Analysis Solution

Archimedean Property

Density Property

Least Upper Bounds Similar for greatest lower bound, and infimum

Equivalent Statement Similar for greatest lower bound, and infimum

Example 4 Show that Analysis Solution

Example 5 Determine the supremum of Analysis Solution

The Completeness Axiom