| Term | Definition |
|---|
| Law of Trichotomy | If x and y are elements of an ordered field F, then exactly one of the relations x<y, x=y, or x>y holds. |
| Supremum | b is the _____ of S if b is an upper bound and moreover b is less than or equal to every other upper bound of S. |
| Upper Bound | Let S be a set in an ordered field F. A number b in F is called ________ for S if for every x in S we have b>=x |
| Infimum | b is the ____ if b is a lower bound for S and if b is greater than or equal to every other lower bound of S. |
| Lower Bound | Let S be a set in an ordered field F. A number b in F is called a ______ for S if for every x in S we have b=<x |
| Axiom of Completeness | Let S be a nonempty set in and ordered field F that has and upper bound. Then F has a least upper bound. Same with lower bounds. |
| Nested Interval Property | Consider a nested sequence of closed sets I_n = [a_n,b_n] in R, that is I_n = {x in R: a_n=< x =<b_n} and I_n is a proper subset of I_n. Then the intersection from n =1..infinity is not empty. |
Combine with other sets
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